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Search: a340785 -id:a340785
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A035363
Number of partitions of n into even parts.
+10
141
1, 0, 1, 0, 2, 0, 3, 0, 5, 0, 7, 0, 11, 0, 15, 0, 22, 0, 30, 0, 42, 0, 56, 0, 77, 0, 101, 0, 135, 0, 176, 0, 231, 0, 297, 0, 385, 0, 490, 0, 627, 0, 792, 0, 1002, 0, 1255, 0, 1575, 0, 1958, 0, 2436, 0, 3010, 0, 3718, 0, 4565, 0, 5604, 0, 6842, 0, 8349, 0, 10143, 0, 12310, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
Convolved with A036469 = A000070. - Gary W. Adamson, Jun 09 2009
Note that these partitions are located in the head of the last section of the set of partitions of n (see A135010). - Omar E. Pol, Nov 20 2009
Number of symmetric unimodal compositions of n+2 where the maximal part appears twice, see example. Also number of symmetric unimodal compositions of n where the maximal part appears an even number of times. - Joerg Arndt, Jun 11 2013
Number of partitions of n having parts of even multiplicity. These are the conjugates of the partitions from the definition. Example: a(8)=5 because we have [4,4],[3,3,1,1],[2,2,2,2],[2,2,1,1,1,1], and [1,1,1,1,1,1,1,1]. - Emeric Deutsch, Jan 27 2016
From Gus Wiseman, May 22 2021: (Start)
The Heinz numbers of the conjugate partitions described in Emeric Deutsch's comment above are given by A000290.
For n > 1, also the number of integer partitions of n-1 whose only odd part is the smallest. The Heinz numbers of these partitions are given by A341446. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns shown as dots, A..D = 10..13) are:
1 . 3 . 5 . 7 . 9 . B . D
21 41 43 63 65 85
221 61 81 83 A3
421 441 A1 C1
2221 621 443 643
4221 641 661
22221 821 841
4421 A21
6221 4441
42221 6421
222221 8221
44221
62221
422221
2222221
Also the number of integer partitions of n whose greatest part is the sum of all the other parts. The Heinz numbers of these partitions are given by A344415. For example, the a(2) = 1 through a(12) = 11 partitions (empty columns not shown) are:
(11) (22) (33) (44) (55) (66)
(211) (321) (422) (532) (633)
(3111) (431) (541) (642)
(4211) (5221) (651)
(41111) (5311) (6222)
(52111) (6321)
(511111) (6411)
(62211)
(63111)
(621111)
(6111111)
Also the number of integer partitions of n of length n/2. The Heinz numbers of these partitions are given by A340387. For example, the a(2) = 1 through a(14) = 15 partitions (empty columns not shown) are:
(2) (22) (222) (2222) (22222) (222222) (2222222)
(31) (321) (3221) (32221) (322221) (3222221)
(411) (3311) (33211) (332211) (3322211)
(4211) (42211) (333111) (3332111)
(5111) (43111) (422211) (4222211)
(52111) (432111) (4322111)
(61111) (441111) (4331111)
(522111) (4421111)
(531111) (5222111)
(621111) (5321111)
(711111) (5411111)
(6221111)
(6311111)
(7211111)
(8111111)
(End)
REFERENCES
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.
LINKS
Robert Price, Table of n, a(n) for n = 0..2001
FORMULA
G.f.: Product_{k even} 1/(1 - x^k).
Convolution with the number of partitions into distinct parts (A000009, which is also number of partitions into odd parts) gives the number of partitions (A000041). - Franklin T. Adams-Watters, Jan 06 2006
If n is even then a(n)=A000041(n/2) otherwise a(n)=0. - Omar E. Pol, Nov 20 2009
G.f.: 1 + x^2*(1 - G(0))/(1-x^2) where G(k) = 1 - 1/(1-x^(2*k+2))/(1-x^2/(x^2-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
a(n) = A096441(n) - A000009(n), n >= 1. - Omar E. Pol, Aug 16 2013
G.f.: exp(Sum_{k>=1} x^(2*k)/(k*(1 - x^(2*k)))). - Ilya Gutkovskiy, Aug 13 2018
EXAMPLE
From Joerg Arndt, Jun 11 2013: (Start)
There are a(12)=11 symmetric unimodal compositions of 12+2=14 where the maximal part appears twice:
01: [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
02: [ 1 1 1 1 3 3 1 1 1 1 ]
03: [ 1 1 1 4 4 1 1 1 ]
04: [ 1 1 2 3 3 2 1 1 ]
05: [ 1 1 5 5 1 1 ]
06: [ 1 2 4 4 2 1 ]
07: [ 1 6 6 1 ]
08: [ 2 2 3 3 2 2 ]
09: [ 2 5 5 2 ]
10: [ 3 4 4 3 ]
11: [ 7 7 ]
There are a(14)=15 symmetric unimodal compositions of 14 where the maximal part appears an even number of times:
01: [ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
02: [ 1 1 1 1 1 2 2 1 1 1 1 1 ]
03: [ 1 1 1 1 3 3 1 1 1 1 ]
04: [ 1 1 1 2 2 2 2 1 1 1 ]
05: [ 1 1 1 4 4 1 1 1 ]
06: [ 1 1 2 3 3 2 1 1 ]
07: [ 1 1 5 5 1 1 ]
08: [ 1 2 2 2 2 2 2 1 ]
09: [ 1 2 4 4 2 1 ]
10: [ 1 3 3 3 3 1 ]
11: [ 1 6 6 1 ]
12: [ 2 2 3 3 2 2 ]
13: [ 2 5 5 2 ]
14: [ 3 4 4 3 ]
15: [ 7 7 ]
(End)
a(8)=5 because we have [8], [6,2], [4,4], [4,2,2], and [2,2,2,2]. - Emeric Deutsch, Jan 27 2016
From Gus Wiseman, May 22 2021: (Start)
The a(0) = 1 through a(12) = 11 partitions into even parts are the following (empty columns shown as dots, A = 10, C = 12). The Heinz numbers of these partitions are given by A066207.
() . (2) . (4) . (6) . (8) . (A) . (C)
(22) (42) (44) (64) (66)
(222) (62) (82) (84)
(422) (442) (A2)
(2222) (622) (444)
(4222) (642)
(22222) (822)
(4422)
(6222)
(42222)
(222222)
(End)
MAPLE
ZL:= [S, {C = Cycle(B), S = Set(C), E = Set(B), B = Prod(Z, Z)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..69); # Zerinvary Lajos, Mar 26 2008
g := 1/mul(1-x^(2*k), k = 1 .. 100): gser := series(g, x = 0, 80): seq(coeff(gser, x, n), n = 0 .. 78); # Emeric Deutsch, Jan 27 2016
# Using the function EULER from Transforms (see link at the bottom of the page).
[1, op(EULER([0, 1, seq(irem(n, 2), n=0..66)]))]; # Peter Luschny, Aug 19 2020
# next Maple program:
a:= n-> `if`(n::odd, 0, combinat[numbpart](n/2)):
seq(a(n), n=0..84); # Alois P. Heinz, Jun 22 2021
MATHEMATICA
nmax = 50; s = Range[2, nmax, 2];
Table[Count[IntegerPartitions@n, x_ /; SubsetQ[s, x]], {n, 0, nmax}] (* Robert Price, Aug 05 2020 *)
PROG
(Python)
from sympy import npartitions
def A035363(n): return 0 if n&1 else npartitions(n>>1) # Chai Wah Wu, Sep 23 2023
CROSSREFS
Bisection (even part) gives the partition numbers A000041.
Column k=0 of A103919, A264398.
Cf. A036469, A000070.
Cf. A135010, A138121.
Note: A-numbers of ranking sequences are in parentheses below.
The version for odd instead of even parts is A000009 (A066208).
The version for parts divisible by 3 instead of 2 is A035377.
The strict case is A035457.
The Heinz numbers of these partitions are given by A066207.
The ordered version (compositions) is A077957 prepended by (1,0).
This is column k = 2 of A168021.
The multiplicative version (factorizations) is A340785.
A000569 counts graphical partitions (A320922).
A004526 counts partitions of length 2 (A001358).
A025065 counts palindromic partitions (A265640).
A027187 counts partitions with even length/maximum (A028260/A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
The following count partitions of even length:
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A000041, A000290, A087897, A100484, A110618, A209816, A210249, A233771, A339004, A340385, A340387, A340786, A341447.
KEYWORD
nonn,easy
AUTHOR
Olivier Gérard
STATUS
approved
A236913
Number of partitions of 2n of type EE (see Comments).
+10
49
1, 1, 3, 6, 12, 22, 40, 69, 118, 195, 317, 505, 793, 1224, 1867, 2811, 4186, 6168, 9005, 13026, 18692, 26613, 37619, 52815, 73680, 102162, 140853, 193144, 263490, 357699, 483338, 650196, 870953, 1161916, 1544048, 2044188, 2696627, 3545015, 4644850, 6066425
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The partitions of n are partitioned into four types:
EO, even # of odd parts and odd # of even parts, A236559;
OE, odd # of odd parts and even # of even parts, A160786;
EE, even # of odd parts and even # of even parts, A236913;
OO, odd # of odd parts and odd # of even parts, A236914.
A236559 and A160786 are the bisections of A027193;
A236913 and A236914 are the bisections of A027187.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
EXAMPLE
The partitions of 4 of type EE are [3,1], [2,2], [1,1,1,1], so that a(2) = 3.
type/k . 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 ... 9 ... 10 .. 11
EO ..... 0 .. 1 .. 0 .. 2 .. 0 .. 5 .. 0 .. 10 .. 0 ... 20 .. 0
OE ..... 1 .. 0 .. 2 .. 0 .. 4 .. 0 .. 8 .. 0 ... 16 .. 0 ... 29
EE ..... 0 .. 1 .. 0 .. 3 .. 0 .. 6 .. 0 .. 12 .. 0 ... 22 .. 0
OO ..... 0 .. 0 .. 1 .. 0 .. 3 .. 0 .. 7 .. 0 ... 14 .. 0 ... 27
From Gus Wiseman, Feb 09 2021: (Start)
This sequence counts even-length partitions of even numbers, which have Heinz numbers given by A340784. For example, the a(0) = 1 through a(4) = 12 partitions are:
() (11) (22) (33) (44)
(31) (42) (53)
(1111) (51) (62)
(2211) (71)
(3111) (2222)
(111111) (3221)
(3311)
(4211)
(5111)
(221111)
(311111)
(11111111)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, [1, 0$3],
`if`(i<1, [0$4], b(n, i-1)+`if`(i>n, [0$4], (p->
`if`(irem(i, 2)=0, [p[3], p[4], p[1], p[2]],
[p[2], p[1], p[4], p[3]]))(b(n-i, i)))))
end:
a:= n-> b(2*n$2)[1]:
seq(a(n), n=0..40); # Alois P. Heinz, Feb 16 2014
MATHEMATICA
z = 25; m1 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &, OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]]; m2 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &, OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]]; m3 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,
OddQ[IntegerPartitions[2 #]]], EvenQ[(*Odd*)First[#]] && EvenQ[(*Even*)Last[#]] &]] &, Range[z]] ; m4 = Map[Length[Select[Map[{Count[#, True], Count[#, False]} &,
OddQ[IntegerPartitions[2 # - 1]]], OddQ[(*Odd*)First[#]] && OddQ[(*Even*)Last[#]] &]] &, Range[z]];
m1 (* A236559, type EO*)
m2 (* A160786, type OE*)
m3 (* A236913, type EE*)
m4 (* A236914, type OO*)
(* Peter J. C. Moses, Feb 03 2014 *)
b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i < 1, {0, 0, 0, 0}, b[n, i - 1] + If[i > n, {0, 0, 0, 0}, Function[p, If[Mod[i, 2] == 0, p[[{3, 4, 1, 2}]], p[[{2, 1, 4, 3}]]]][b[n - i, i]]]]]; a[n_] := b[2*n, 2*n][[1]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 27 2015, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[2n], EvenQ[Length[#]]&]], {n, 0, 15}] (* Gus Wiseman, Feb 09 2021 *)
CROSSREFS
Cf. A000041, A027193, A236559, A236914.
Note: A-numbers of ranking sequences are in parentheses below.
The ordered version is A000302.
The case of odd-length partitions of odd numbers is A160786 (A340931).
The Heinz numbers of these partitions are (A340784).
A027187 counts partitions of even length/maximum (A028260/A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A047993 counts balanced partitions (A106529).
A058695 counts partitions of odd numbers (A300063).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A072233 counts partitions by sum and length.
A339846 counts factorizations of even length.
A340601 counts partitions of even rank (A340602).
A340785 counts factorizations into even factors.
A340786 counts even-length factorizations into even factors.
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 01 2014
EXTENSIONS
More terms from Alois P. Heinz, Feb 16 2014
STATUS
approved
A096441
Number of palindromic and unimodal compositions of n. Equivalently, the number of orbits under conjugation of even nilpotent n X n matrices.
+10
29
1, 2, 2, 4, 3, 7, 5, 11, 8, 17, 12, 26, 18, 37, 27, 54, 38, 76, 54, 106, 76, 145, 104, 199, 142, 266, 192, 357, 256, 472, 340, 621, 448, 809, 585, 1053, 760, 1354, 982, 1740, 1260, 2218, 1610, 2818, 2048, 3559, 2590, 4485, 3264, 5616, 4097, 7018, 5120, 8728, 6378
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Number of partitions of n such that all differences between successive parts are even, see example. [Joerg Arndt, Dec 27 2012]
Number of partitions of n where either all parts are odd or all parts are even. - Omar E. Pol, Aug 16 2013
From Gus Wiseman, Jan 13 2022: (Start)
Also the number of integer partitions of n with all even multiplicities (or run-lengths) except possibly the first. These are the conjugates of the partitions described by Joerg Arndt above. For example, the a(1) = 1 through a(8) = 11 partitions are:
(1) (2) (3) (4) (5) (6) (7) (8)
(11) (111) (22) (311) (33) (322) (44)
(211) (11111) (222) (511) (422)
(1111) (411) (31111) (611)
(2211) (1111111) (2222)
(21111) (3311)
(111111) (22211)
(41111)
(221111)
(2111111)
(11111111)
(End)
REFERENCES
A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras. Lie groups and invariant theory, 85-104, Amer. Math. Soc. Transl. Ser. 2, 213, Amer. Math. Soc., Providence, RI, 2005.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000
Karin Baur and Nolan Wallach, Nice parabolic subalgebras of reductive Lie algebras, Represent. Theory 9 (2005), 1-29.
A. G. Elashvili and V. G. Kac, Classification of good gradings of simple Lie algebras, arXiv:math-ph/0312030, 2002-2004.
FORMULA
G.f.: sum(j>=1, q^j * (1-q^j)/prod(i=1..j, 1-q^(2*i) ) ).
G.f.: F + G - 2, where F = prod(j>=1, 1/(1-q^(2*j) ), G = prod(j>=0, 1/(1-q^(2*j+1)) ).
a(2*n) = A000041(n) + A000009(2*n); a(2*n-1) = A000009(2*n-1). - Vladeta Jovovic, Aug 11 2004
a(n) = A000009(n) + A035363(n) = A000041(n) - A006477(n). - Omar E. Pol, Aug 16 2013
EXAMPLE
From Joerg Arndt, Dec 27 2012: (Start)
There are a(10)=17 partitions of 10 where all differences between successive parts are even:
[ 1] [ 1 1 1 1 1 1 1 1 1 1 ]
[ 2] [ 2 2 2 2 2 ]
[ 3] [ 3 1 1 1 1 1 1 1 ]
[ 4] [ 3 3 1 1 1 1 ]
[ 5] [ 3 3 3 1 ]
[ 6] [ 4 2 2 2 ]
[ 7] [ 4 4 2 ]
[ 8] [ 5 1 1 1 1 1 ]
[ 9] [ 5 3 1 1 ]
[10] [ 5 5 ]
[11] [ 6 2 2 ]
[12] [ 6 4 ]
[13] [ 7 1 1 1 ]
[14] [ 7 3 ]
[15] [ 8 2 ]
[16] [ 9 1 ]
[17] [ 10 ]
(End)
MAPLE
b:= proc(n, i) option remember; `if`(i>n, 0,
`if`(irem(n, i)=0, 1, 0) +add(`if`(irem(j, 2)=0,
b(n-i*j, i+1), 0), j=0..n/i))
end:
a:= n-> b(n, 1):
seq(a(n), n=1..60); # Alois P. Heinz, Mar 26 2014
MATHEMATICA
(* The following Mathematica program first generates all of the palindromic, unimodal compositions of n and then counts them. *)
Pal[n_] := Block[{i, j, k, m, Q, L}, If[n == 1, Return[{{1}}]]; If[n == 2, Return[{{1, 1}, {2}}]]; L = {{n}}; If[Mod[n, 2] == 0, L = Append[L, {n/2, n/2}]]; For[i = 1, i < n, i++, Q = Pal[n - 2i]; m = Length[Q]; For[j = 1, j <= m, j++, If[i <= Q[[j, 1]], L = Append[L, Append[Prepend[Q[[j]], i], i]]]]]; L] NoPal[n_] := Length[Pal[n]]
a[n_] := PartitionsQ[n] + If[EvenQ[n], PartitionsP[n/2], 0]; Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Mar 17 2014, after Vladeta Jovovic *)
Table[Length[Select[IntegerPartitions[n], And@@EvenQ/@Rest[Length/@Split[#]]&]], {n, 1, 30}] (* Gus Wiseman, Jan 13 2022 *)
PROG
(PARI) x='x+O('x^66); Vec(eta(x^2)/eta(x)+1/eta(x^2)-2) \\ Joerg Arndt, Jan 17 2016
CROSSREFS
Bisections are A078408 and A096967.
The complement in partitions is counted by A006477
A version for compositions is A016116.
A pointed version is A035363, ranked by A066207.
A000041 counts integer partitions.
A025065 counts palindromic partitions.
A027187 counts partitions with even length/maximum.
A035377 counts partitions using multiples of 3.
A058696 counts partitions of even numbers, ranked by A300061.
A340785 counts factorizations into even factors.
Cf. A000009, A002865, A027383, A035457, A117298, A117989, A168021, A274230, A345170, A349060, A349061.
KEYWORD
nonn
AUTHOR
Nolan R. Wallach (nwallach(AT)ucsd.edu), Aug 10 2004
STATUS
approved
A340101
Number of factorizations of 2n + 1 into odd factors > 1.
+10
28
1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 4, 1, 2, 2, 1, 2, 2, 1, 1, 4, 2, 1, 2, 1, 1, 4, 2, 1, 5, 1, 2, 2, 1, 2, 2, 2, 1, 4, 1, 1, 5, 1, 1, 2, 1, 2, 4, 2, 2, 2, 3, 1, 2, 1, 2, 7, 1, 1, 2, 2, 2, 4, 1, 1, 4, 2, 1, 2, 2, 1, 5, 1, 2, 4, 1, 4, 2, 1, 1, 2, 2, 2, 7, 1, 1, 5, 1, 1, 2, 2, 2, 4, 2
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
LINKS
Antti Karttunen, Table of n, a(n) for n = 0..32768
FORMULA
a(n) = A001055(2n+1).
a(n) = A349907(2n+1). - Antti Karttunen, Dec 13 2021
EXAMPLE
The factorizations for 2n + 1 = 27, 45, 135, 225, 315, 405, 1155:
27 45 135 225 315 405 1155
3*9 5*9 3*45 3*75 5*63 5*81 15*77
3*3*3 3*15 5*27 5*45 7*45 9*45 21*55
3*3*5 9*15 9*25 9*35 15*27 33*35
3*5*9 15*15 15*21 3*135 3*385
3*3*15 5*5*9 3*105 5*9*9 5*231
3*3*3*5 3*3*25 5*7*9 3*3*45 7*165
3*5*15 3*3*35 3*5*27 11*105
3*3*5*5 3*5*21 3*9*15 3*5*77
3*7*15 3*3*5*9 3*7*55
3*3*5*7 3*3*3*15 5*7*33
3*3*3*3*5 3*11*35
5*11*21
7*11*15
3*5*7*11
MAPLE
g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
`if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
d=numtheory[divisors](n) minus {1, n}))
end:
a:= n-> g(2*n+1$2):
seq(a(n), n=0..100); # Alois P. Heinz, Dec 30 2020
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], OddQ[Times@@#]&]], {n, 1, 100, 2}]
PROG
(PARI)
A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s)); \\ After code in A001055
A340101(n) = A001055(n+n+1); \\ Antti Karttunen, Dec 13 2021
CROSSREFS
The version for partitions is A160786, ranked by A300272.
The even version is A340785.
The odd-length case is A340102.
A000009 counts partitions into odd parts, ranked by A066208.
A001055 counts factorizations, with strict case A045778.
A027193 counts partitions of odd length, ranked by A026424.
A058695 counts partitions of odd numbers, ranked by A300063.
A316439 counts factorizations by product and length.
Cf. A000700, A002033, A027187, A028260, A074206, A078408, A174726, A236914, A320732, A339846.
Odd bisection of A001055, and also of A349907.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 28 2020
EXTENSIONS
Data section extended up to 105 terms by Antti Karttunen, Dec 13 2021
STATUS
approved
A340784
Heinz numbers of even-length integer partitions of even numbers.
+10
16
1, 4, 9, 10, 16, 21, 22, 25, 34, 36, 39, 40, 46, 49, 55, 57, 62, 64, 81, 82, 84, 85, 87, 88, 90, 91, 94, 100, 111, 115, 118, 121, 129, 133, 134, 136, 144, 146, 155, 156, 159, 160, 166, 169, 183, 184, 187, 189, 194, 196, 198, 203, 205, 206, 210, 213, 218, 220
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are positive integers whose number of prime indices and sum of prime indices are both even, counting multiplicity in both cases.
A multiplicative semigroup: if m and n are in the sequence, then so is m*n. - Antti Karttunen, Jul 28 2024
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..20000
Index entries for sequences computed from indices in prime factorization
FORMULA
Intersection of A028260 and A300061.
EXAMPLE
The sequence of partitions together with their Heinz numbers begins:
1: () 57: (8,2) 118: (17,1)
4: (1,1) 62: (11,1) 121: (5,5)
9: (2,2) 64: (1,1,1,1,1,1) 129: (14,2)
10: (3,1) 81: (2,2,2,2) 133: (8,4)
16: (1,1,1,1) 82: (13,1) 134: (19,1)
21: (4,2) 84: (4,2,1,1) 136: (7,1,1,1)
22: (5,1) 85: (7,3) 144: (2,2,1,1,1,1)
25: (3,3) 87: (10,2) 146: (21,1)
34: (7,1) 88: (5,1,1,1) 155: (11,3)
36: (2,2,1,1) 90: (3,2,2,1) 156: (6,2,1,1)
39: (6,2) 91: (6,4) 159: (16,2)
40: (3,1,1,1) 94: (15,1) 160: (3,1,1,1,1,1)
46: (9,1) 100: (3,3,1,1) 166: (23,1)
49: (4,4) 111: (12,2) 169: (6,6)
55: (5,3) 115: (9,3) 183: (18,2)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], EvenQ[PrimeOmega[#]]&&EvenQ[Total[primeMS[#]]]&]
PROG
(PARI)
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1]))); }
A353331(n) = ((!(bigomega(n)%2)) && (!(A056239(n)%2)));
isA340784(n) = A353331(n); \\ Antti Karttunen, Apr 14 2022
CROSSREFS
Note: A-numbers of Heinz-number sequences are in parentheses below.
The case of prime powers is A056798.
These partitions are counted by A236913.
The odd version is A160786 (A340931).
A000009 counts partitions into odd parts (A066208).
A001222 counts prime factors.
A047993 counts balanced partitions (A106529).
A056239 adds up prime indices.
A058695 counts partitions of odd numbers (A300063).
A061395 selects the maximum prime index.
A072233 counts partitions by sum and length.
A112798 lists the prime indices of each positive integer.
- Even -
A027187 counts partitions of even length/maximum (A028260/A244990).
A034008 counts compositions of even length.
A035363 counts partitions into even parts (A066207).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A339846 counts factorizations of even length.
A340601 counts partitions of even rank (A340602).
A340785 counts factorizations into even factors.
A340786 counts even-length factorizations into even factors.
Cf. A026424, A257541, A300272, A326837, A326845, A340385 (A340386), A340604, A353331 (characteristic function), A353332, A353333, A353334.
Squares (A000290) is a subsequence.
Not a subsequence of A329609 (30 is the first term of A329609 not occurring here, and 210 is the first term here not present in A329609).
Positions of even terms in A373381.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 30 2021
STATUS
approved
A340852
Numbers that can be factored in such a way that every factor is a divisor of the number of factors.
+10
14
1, 4, 16, 27, 32, 64, 96, 128, 144, 192, 216, 256, 288, 324, 432, 486, 512, 576, 648, 729, 864, 972, 1024, 1296, 1458, 1728, 1944, 2048, 2560, 2592, 2916, 3125, 3888, 4096, 5120, 5184, 5832, 6144, 6400, 7776, 8192, 9216, 11664, 12288, 12800, 13824, 15552
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
Also numbers that can be factored in such a way that the length is divisible by the least common multiple.
LINKS
Table of n, a(n) for n=1..47.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
4: {1,1}
16: {1,1,1,1}
27: {2,2,2}
32: {1,1,1,1,1}
64: {1,1,1,1,1,1}
96: {1,1,1,1,1,2}
128: {1,1,1,1,1,1,1}
144: {1,1,1,1,2,2}
192: {1,1,1,1,1,1,2}
216: {1,1,1,2,2,2}
256: {1,1,1,1,1,1,1,1}
288: {1,1,1,1,1,2,2}
324: {1,1,2,2,2,2}
432: {1,1,1,1,2,2,2}
For example, 24576 has three suitable factorizations:
(2*2*2*2*2*2*2*2*2*2*2*12)
(2*2*2*2*2*2*2*2*2*2*4*6)
(2*2*2*2*2*2*2*2*2*3*4*4)
so is in the sequence.
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[1000], Select[facs[#], And@@IntegerQ/@(Length[#]/#)&]!={}&]
CROSSREFS
Partitions of this type are counted by A340693 (A340606).
These factorizations are counted by A340851.
The reciprocal version is A340853.
A143773 counts partitions whose parts are multiples of the number of parts.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
A340656 lack a twice-balanced factorization, complement A340657.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340831/A340832 count factorizations with odd maximum/minimum.
A340785 counts factorizations into even numbers, even-length case A340786.
A340854 cannot be factored with odd least factor, complement A340855.
Cf. A050320, A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340609, A340654, A340655, A340827, A340830.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2021
STATUS
approved
A340831
Number of factorizations of n into factors > 1 with odd greatest factor.
+10
10
0, 0, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 1, 2, 2, 1, 1, 1, 2, 1, 3, 2, 1, 2, 1, 0, 2, 1, 2, 3, 1, 1, 2, 2, 1, 3, 1, 2, 4, 1, 1, 1, 2, 2, 2, 2, 1, 4, 2, 2, 2, 1, 1, 4, 1, 1, 4, 0, 2, 3, 1, 2, 2, 2, 1, 4, 1, 1, 4, 2, 2, 3, 1, 3, 5, 1, 1, 5, 2, 1, 2, 3, 1, 5, 2, 2, 2, 1, 2, 1, 1, 2, 4, 4, 1, 3, 1, 3, 5, 1, 1, 6
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,9
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..20000
EXAMPLE
The a(n) factorizations for n = 45, 108, 135, 180, 252:
(45) (4*27) (135) (4*45) (4*63)
(5*9) (2*6*9) (3*45) (12*15) (12*21)
(3*15) (3*4*9) (5*27) (4*5*9) (4*7*9)
(3*3*5) (2*2*27) (9*15) (2*2*45) (6*6*7)
(2*2*3*9) (3*5*9) (2*6*15) (2*2*63)
(2*2*3*3*3) (3*3*15) (3*4*15) (2*6*21)
(3*3*3*5) (2*2*5*9) (3*4*21)
(3*3*4*5) (2*2*7*9)
(2*2*3*15) (2*3*6*7)
(2*2*3*3*5) (3*3*4*7)
(2*2*3*21)
(2*2*3*3*7)
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], OddQ@*Max]], {n, 100}]
PROG
(PARI) A340831(n, m=n, fc=1) = if(1==n, !fc, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&(!fc||(d%2)), s += A340831(n/d, d, 0*fc))); (s)); \\ Antti Karttunen, Dec 13 2021
CROSSREFS
Positions of 0's are A000079.
The version for partitions is A027193.
The version for prime indices is A244991.
The version looking at length instead of greatest factor is A339890.
The version that also has odd length is A340607.
The version looking at least factor is A340832.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A058695 counts partitions of odd numbers.
A066208 lists numbers with odd-indexed prime factors.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A050320, A061395, A339846, A340385, A340596, A340599, A340654, A340655, A340785, A340854, A340855.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2021
EXTENSIONS
Data section extended up to 108 terms by Antti Karttunen, Dec 13 2021
STATUS
approved
A340786
Number of factorizations of 4n into an even number of even factors > 1.
+10
8
1, 1, 1, 3, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 6, 1, 3, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 4, 1, 7, 2, 2, 2, 7, 1, 2, 2, 6, 1, 4, 1, 4, 3, 2, 1, 10, 2, 3, 2, 4, 1, 4, 2, 6, 2, 2, 1, 8, 1, 2, 3, 12, 2, 4, 1, 4, 2, 4, 1, 10, 1, 2, 3, 4, 2, 4, 1, 10, 3, 2, 1, 8, 2, 2
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
EXAMPLE
The a(n) factorizations for n = 6, 12, 24, 36, 60, 80, 500:
4*6 6*8 2*48 2*72 4*60 4*80 40*50
2*12 2*24 4*24 4*36 6*40 8*40 4*500
4*12 6*16 6*24 8*30 10*32 8*250
2*2*2*6 8*12 8*18 10*24 16*20 10*200
2*2*4*6 12*12 12*20 2*160 20*100
2*2*2*12 2*2*6*6 2*120 2*2*2*40 2*1000
2*2*2*18 2*2*2*30 2*2*4*20 2*2*10*50
2*2*6*10 2*2*8*10 2*2*2*250
2*4*4*10 2*10*10*10
2*2*2*2*2*10
MAPLE
g:= proc(n, m, p)
option remember;
local F, r, x, i;
# number of factorizations of n into even factors > m with number of factors == p (mod 2)
if n = 1 then if p = 0 then return 1 else return 0 fi fi;
if m > n or n::odd then return 0 fi;
F:= sort(convert(select(t -> t > m and t::even, numtheory:-divisors(n)), list));
r:= 0;
for x in F do
for i from 1 while n mod x^i = 0 do
r:= r + procname(n/x^i, x, (p+i) mod 2)
od od;
r
end proc:
f:= n -> g(4*n, 1, 0):
map(f, [$1..100]); # Robert Israel, Mar 16 2023
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[4n], EvenQ[Length[#]]&&Select[#, OddQ]=={}&]], {n, 100}]
PROG
(PARI)
A340786aux(n, m=n, p=0) = if(1==n, (0==p), my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&!(d%2), s += A340786aux(n/d, d, 1-p))); (s));
A340786(n) = A340786aux(4*n); \\ Antti Karttunen, Apr 14 2022
CROSSREFS
Note: A-numbers of Heinz-number sequences are in parentheses below.
Positions of ones are 1 and A000040, or A008578.
A version for partitions is A027187 (A028260).
Allowing odd length gives A108501 (bisection of A340785).
Allowing odd factors gives A339846.
An odd version is A340102.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors.
A340653 counts balanced factorizations.
A340831/A340832 count factorizations with odd maximum/minimum.
- Even -
A027187 counts partitions of even maximum (A244990).
A058696 counts partitions of even numbers (A300061).
A067661 counts strict partitions of even length (A030229).
A236913 counts partitions of even length and sum (A340784).
A340601 counts partitions of even rank (A340602).
Cf. A001147, A001222, A001749, A035363, A050320, A066207, A066208, A160786, A174725, A320655, A320656, A339890, A340851.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 31 2021
STATUS
approved
A340851
Number of factorizations of n such that every factor is a divisor of the number of factors.
+10
8
1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,64
COMMENTS
Also factorizations whose number of factors is divisible by their least common multiple.
LINKS
Table of n, a(n) for n=1..87.
EXAMPLE
The a(n) factorizations for n = 8192, 46656, 73728:
2*2*2*2*2*4*8*8 6*6*6*6*6*6 2*2*2*2*2*2*2*2*2*4*6*6
2*2*2*2*4*4*4*8 2*2*2*2*2*2*3*3*3*3*3*3 2*2*2*2*2*2*2*2*3*4*4*6
2*2*2*4*4*4*4*4 2*2*2*2*2*2*2*3*3*4*4*4
2*2*2*2*2*2*2*2*2*2*2*4 2*2*2*2*2*2*2*2*2*2*6*12
2*2*2*2*2*2*2*2*2*3*4*12
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], And@@IntegerQ/@(Length[#]/#)&]], {n, 100}]
CROSSREFS
The version for partitions is A340693, with reciprocal version A143773.
Positions of nonzero terms are A340852.
The reciprocal version is A340853.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
A340656 lack a twice-balanced factorization, complement A340657.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340785 counts factorizations into even numbers, even-length case A340786.
A340831/A340832 count factorizations with odd maximum/minimum.
A340854 cannot be factored with odd least factor, complement A340855.
Cf. A067538, A074761, A168659, A301987, A327517, A340596, A340599, A340654, A340655, A340827, A340830.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2021
STATUS
approved
A340853
Number of factorizations of n such that every factor is a multiple of the number of factors.
+10
8
0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 1, 3, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 4, 2, 1, 1, 3, 1, 1, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
Also factorizations whose greatest common divisor is a multiple of the number of factors.
LINKS
Table of n, a(n) for n=1..87.
EXAMPLE
The a(n) factorizations for n = 2, 4, 16, 48, 96, 144, 216, 240, 432:
2 4 16 48 96 144 216 240 432
2*2 2*8 6*8 2*48 2*72 4*54 4*60 6*72
4*4 2*24 4*24 4*36 6*36 6*40 8*54
4*12 6*16 6*24 12*18 8*30 12*36
8*12 8*18 2*108 10*24 18*24
12*12 6*6*6 12*20 2*216
3*3*24 2*120 4*108
3*6*12 3*3*48
3*6*24
6*6*12
3*12*12
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Select[facs[n], n>1&&Divisible[GCD@@#, Length[#]]&]], {n, 100}]
CROSSREFS
Positions of 1's are A048103.
Positions of terms > 1 are A100716.
The version for partitions is A143773 (A316428).
The reciprocal for partitions is A340693 (A340606).
The version for strict partitions is A340830.
The reciprocal version is A340851.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
A340656 lack a twice-balanced factorization, complement A340657.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340785 counts factorizations into even factors, even-length case A340786.
A340831/A340832 counts factorizations with odd maximum/minimum.
A340854 cannot be factored with odd least factor, complement A340855.
Cf. A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340654, A340655, A340827.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2021
STATUS
approved
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