Displaying 1-10 of 21 results found.
Triangle read by rows where T(n,k) is the number of integer partitions of n whose left half (exclusive) sums to k, where k ranges from 0 to n.
+10
27
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 3, 1, 0, 1, 0, 2, 3, 1, 0, 1, 0, 1, 4, 4, 1, 0, 1, 0, 0, 3, 6, 4, 1, 0, 1, 0, 0, 1, 7, 7, 5, 1, 0, 1, 0, 0, 1, 4, 8, 10, 5, 1, 0, 1, 0, 0, 0, 3, 6, 14, 11, 6, 1, 0, 1, 0, 0, 0, 1, 5, 12, 16, 14, 6, 1, 0
COMMENTS
Also the number of integer partitions of n whose right half (inclusive) sums to n-k.
EXAMPLE
Triangle begins:
1
1 0
1 1 0
1 1 1 0
1 0 3 1 0
1 0 2 3 1 0
1 0 1 4 4 1 0
1 0 0 3 6 4 1 0
1 0 0 1 7 7 5 1 0
1 0 0 1 4 8 10 5 1 0
1 0 0 0 3 6 14 11 6 1 0
1 0 0 0 1 5 12 16 14 6 1 0
1 0 0 0 1 2 12 14 23 16 7 1 0
1 0 0 0 0 2 7 13 24 27 19 7 1 0
1 0 0 0 0 1 5 9 24 30 35 21 8 1 0
1 0 0 0 0 1 3 7 17 31 42 40 25 8 1 0
1 0 0 0 0 0 2 4 16 23 46 51 51 27 9 1 0
1 0 0 0 0 0 1 3 10 21 37 57 69 57 31 9 1 0
1 0 0 0 0 0 1 2 7 15 34 47 83 81 69 34 10 1 0
For example, row n = 9 counts the following partitions:
(9) . . (333) (432) (54) (63) (72) (81)
(441) (522) (621) (711)
(22221) (531) (3321) (4311)
(111111111) (3222) (4221) (5211)
(32211) (33111) (6111)
(2211111) (42111)
(3111111) (51111)
(21111111) (222111)
(321111)
(411111)
For example, the partition y = (3,2,2,1,1) has left half (exclusive) (3,2), with sum 5, so y is counted under T(9,5).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Total[Take[#, Floor[Length[#]/2]]]==k&]], {n, 0, 10}, {k, 0, n}]
CROSSREFS
The left inclusive version is A360675 with rows reversed. A008284 counts partitions by length.
First for prime indices, second for partitions, third for prime factors:
Triangle read by rows where T(n,k) is the number of integer partitions of n whose right half (exclusive) sums to k, where k ranges from 0 to n.
+10
27
1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 0, 1, 3, 3, 0, 0, 0, 1, 3, 5, 2, 0, 0, 0, 1, 4, 6, 4, 0, 0, 0, 0, 1, 4, 9, 5, 3, 0, 0, 0, 0, 1, 5, 10, 10, 4, 0, 0, 0, 0, 0, 1, 5, 13, 12, 9, 2, 0, 0, 0, 0, 0, 1, 6, 15, 18, 11, 5, 0, 0, 0, 0, 0, 0
COMMENTS
Also the number of integer partitions of n whose left half (inclusive) sums to n-k.
EXAMPLE
Triangle begins:
1
1 0
1 1 0
1 2 0 0
1 2 2 0 0
1 3 3 0 0 0
1 3 5 2 0 0 0
1 4 6 4 0 0 0 0
1 4 9 5 3 0 0 0 0
1 5 10 10 4 0 0 0 0 0
1 5 13 12 9 2 0 0 0 0 0
1 6 15 18 11 5 0 0 0 0 0 0
1 6 18 22 20 6 4 0 0 0 0 0 0
1 7 20 29 26 13 5 0 0 0 0 0 0 0
1 7 24 34 37 19 11 2 0 0 0 0 0 0 0
1 8 26 44 46 30 16 5 0 0 0 0 0 0 0 0
1 8 30 50 63 40 27 8 4 0 0 0 0 0 0 0 0
1 9 33 61 75 61 36 15 6 0 0 0 0 0 0 0 0 0
1 9 37 70 96 75 61 21 12 3 0 0 0 0 0 0 0 0 0
For example, row n = 9 counts the following partitions:
(9) (81) (72) (63) (54)
(441) (432) (333) (3222)
(531) (522) (3321) (21111111)
(621) (4311) (4221) (111111111)
(711) (5211) (22221)
(6111) (222111)
(32211) (321111)
(33111) (411111)
(42111) (2211111)
(51111) (3111111)
For example, the partition y = (3,2,2,1,1) has right half (exclusive) (1,1), with sum 2, so y is counted under T(9,2).
MATHEMATICA
Table[Length[Select[IntegerPartitions[n], Total[Take[#, -Floor[Length[#]/2]]]==k&]], {n, 0, 18}, {k, 0, n}]
CROSSREFS
The central diagonal T(2n,n) is A000005. The right inclusive version is A360672 with rows reversed. A008284 counts partitions by length.
First for prime indices, second for partitions, third for prime factors:
Number of divisors of n with exactly half as many prime factors as n, counting multiplicity.
+10
22
1, 0, 0, 1, 0, 2, 0, 0, 1, 2, 0, 0, 0, 2, 2, 1, 0, 0, 0, 0, 2, 2, 0, 2, 1, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, 3, 0, 2, 2, 2, 0, 0, 0, 0, 0, 2, 0, 0, 1, 0, 2, 0, 0, 2, 2, 2, 2, 2, 0, 4, 0, 2, 0, 1, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 1, 2, 0, 4, 2, 2, 2
COMMENTS
These divisors do not necessarily include the central divisors ( A207375), and may not themselves be central.
EXAMPLE
The a(n) divisors for selected n:
n = 1: 6: 36: 60: 210: 840: 900: 1260: 1296: 3600:
--------------------------------------------------------
1 2 4 4 6 8 12 12 16 16
3 6 6 10 12 18 18 24 24
9 10 14 20 20 20 36 36
15 15 28 30 28 54 40
21 30 45 30 81 60
35 42 50 42 90
70 75 45 100
105 63 150
70 225
105
MATHEMATICA
Table[Length[Select[Divisors[n], PrimeOmega[#]==PrimeOmega[n]/2&]], {n, 100}]
PROG
(PARI) a(n) = my(nb=bigomega(n)); sumdiv(n, d, 2*bigomega(d) == nb); \\ Michel Marcus, Aug 16 2021 (Python)
from sympy import divisors, factorint
def a(n):
npf = len(factorint(n, multiple=True))
divs = divisors(n)
return sum(2*len(factorint(d, multiple=True)) == npf for d in divs)
(Python 3.8+)
from itertools import combinations
from math import prod, comb
from sympy import factorint
if n == 1:
return 1
fs = factorint(n)
elist = list(fs.values())
q, r = divmod(sum(elist), 2)
k = len(elist)
if r:
return 0
c = 0
for i in range(k+1):
m = (-1)**i
for d in combinations(range(k), i):
t = k+q-sum(elist[j] for j in d)-i-1
if t >= 0:
c += m*comb(t, k-1)
(Python)
from sympy import factorint
from sympy.utilities.iterables import multiset_combinations
if n == 1:
return 1
fs = factorint(n, multiple=True)
q, r = divmod(len(fs), 2)
return 0 if r else len(list(multiset_combinations(fs, q))) # Chai Wah Wu, Aug 20 2021
CROSSREFS
The case of powers of 2 is A000035. Positions of even terms are A000037. Positions of odd terms are A000290. The case of all divisors (not just 2) is A347042. A001221 counts distinct prime factors.
A334997 counts chains of divisors of n by length.
Smallest divisor of n with half (rounded up) as many prime factors (counting multiplicity) as n.
+10
21
1, 2, 3, 2, 5, 2, 7, 4, 3, 2, 11, 4, 13, 2, 3, 4, 17, 6, 19, 4, 3, 2, 23, 4, 5, 2, 9, 4, 29, 6, 31, 8, 3, 2, 5, 4, 37, 2, 3, 4, 41, 6, 43, 4, 9, 2, 47, 8, 7, 10, 3, 4, 53, 6, 5, 4, 3, 2, 59, 4, 61, 2, 9, 8, 5, 6, 67, 4, 3, 10, 71, 8, 73, 2, 15, 4, 7, 6, 79, 8
COMMENTS
Appears to contain every positive integer at least once.
EXAMPLE
The divisors of 123456 with half bigomega are: 16, 24, 5144, 7716, so a(123456) = 16.
MATHEMATICA
Table[Min[Select[Divisors[n], PrimeOmega[#]==Ceiling[PrimeOmega[n]/2]&]], {n, 100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]}, Times @@ p[[1 ;; Ceiling[Length[p]/2]]]]; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)
PROG
(PARI) a(n) = my(bn=ceil(bigomega(n)/2)); fordiv(n, d, if (bigomega(d)==bn, return (d))); \\ Michel Marcus, Aug 18 2021 (Python)
from sympy import divisors, factorint
def a(n):
npf = len(factorint(n, multiple=True))
for d in divisors(n):
if len(factorint(d, multiple=True)) == (npf+1)//2: return d
return 1
(Python 3.8+)
from math import prod
from sympy import factorint
fs = factorint(n, multiple=True)
l = len(fs)
CROSSREFS
Positions of odd terms are A005408. Positions of even terms are A005843. The case of powers of 2 is A016116. The smallest divisor without the condition is A020639 (greatest: A006530). A001221 counts distinct prime factors.
A001222 counts all prime factors (also called bigomega).
A340387 lists numbers whose sum of prime indices is twice bigomega.
A340609 lists numbers whose maximum prime index divides bigomega.
A340610 lists numbers whose maximum prime index is divisible by bigomega.
A347042 counts divisors d|n such that bigomega(d) divides bigomega(n).
Number of multisets whose right half (inclusive) sums to n.
+10
17
1, 2, 5, 8, 16, 21, 42, 51, 90, 121, 185, 235, 386, 465, 679, 908, 1261, 1580, 2238, 2770, 3827, 4831, 6314, 7910, 10619, 13074, 16813, 21049, 26934, 33072, 42445, 51679, 65264, 79902, 99309, 121548, 151325, 182697, 224873, 272625, 334536, 401999, 491560, 588723
FORMULA
G.f.: 1 + Sum_{k>=1} x^k*(2 - x^k)/((1 - x^k)^(k+1) * Product_{j=1..k-1} (1-x^j)). - Andrew Howroyd, Mar 11 2023
EXAMPLE
The a(0) = 1 through a(4) = 16 multisets:
{} {1} {2} {3} {4}
{1,1} {1,2} {1,3} {1,4}
{2,2} {2,3} {2,4}
{1,1,1} {3,3} {3,4}
{1,1,1,1} {1,1,2} {4,4}
{1,1,1,2} {1,1,3}
{1,1,1,1,1} {1,2,2}
{1,1,1,1,1,1} {2,2,2}
{1,1,1,3}
{1,1,2,2}
{1,2,2,2}
{2,2,2,2}
{1,1,1,1,2}
{1,1,1,1,1,2}
{1,1,1,1,1,1,1}
{1,1,1,1,1,1,1,1}
For example, the multiset y = {1,1,1,1,2} has right half (inclusive) {1,1,2}, with sum 4, so y is counted under a(4).
MATHEMATICA
Table[Length[Select[Join@@IntegerPartitions/@Range[0, 3*k], Total[Take[#, Ceiling[Length[#]/2]]]==k&]], {k, 0, 15}]
PROG
(PARI) seq(n)={my(s=1 + O(x*x^n), p=s); for(k=1, n, s += p*x^k*(2-x^k)/(1-x^k + O(x*x^(n-k)))^(k+1); p /= 1 - x^k); Vec(s)} \\ Andrew Howroyd, Mar 11 2023
CROSSREFS
Column sums of A360675 with rows reversed. A360672 counts partitions by left sum (exclusive).
A360679 gives right sum (inclusive) of prime indices.
Number of multisets of positive integers whose right half (exclusive) sums to n.
+10
17
1, 2, 7, 13, 27, 37, 73, 89, 156, 205, 315, 387, 644, 749, 1104, 1442, 2015, 2453, 3529, 4239, 5926, 7360, 9624, 11842, 16115, 19445, 25084, 31137, 39911, 48374, 62559, 75135, 95263, 115763, 143749, 174874, 218614, 261419, 321991, 388712, 477439, 569968, 698493
FORMULA
G.f.: 1 + Sum_{k>=1} x^k*(2 - x^k)/((1 - x^k)^(k+2) * Product_{j=1..k-1} (1-x^j)). - Andrew Howroyd, Mar 11 2023
EXAMPLE
The a(0) = 1 through a(3) = 13 multisets:
{} {1,1} {1,2} {1,3}
{1,1,1} {2,2} {2,3}
{1,1,2} {3,3}
{1,2,2} {1,1,3}
{2,2,2} {1,2,3}
{1,1,1,1} {1,3,3}
{1,1,1,1,1} {2,2,3}
{2,3,3}
{3,3,3}
{1,1,1,2}
{1,1,1,1,2}
{1,1,1,1,1,1}
{1,1,1,1,1,1,1}
For example, the multiset y = {1,1,1,1,2} has right half (exclusive) {1,2}, with sum 3, so y is counted under a(3).
MATHEMATICA
Table[Length[Select[Join@@IntegerPartitions/@Range[0, 3*k], Total[Take[#, Floor[Length[#]/2]]]==k&]], {k, 0, 15}]
PROG
(PARI) seq(n)={my(s=1 + O(x*x^n), p=s); for(k=1, n, s += p*x^k*(2-x^k)/(1-x^k + O(x*x^(n-k)))^(k+2); p /= 1 - x^k); Vec(s)} \\ Andrew Howroyd, Mar 11 2023
CROSSREFS
First for prime indices, second for partitions, third for prime factors:
Sum of the right half (inclusive) of the prime indices of n.
+10
17
0, 1, 2, 1, 3, 2, 4, 2, 2, 3, 5, 3, 6, 4, 3, 2, 7, 4, 8, 4, 4, 5, 9, 3, 3, 6, 4, 5, 10, 5, 11, 3, 5, 7, 4, 4, 12, 8, 6, 4, 13, 6, 14, 6, 5, 9, 15, 4, 4, 6, 7, 7, 16, 4, 5, 5, 8, 10, 17, 5, 18, 11, 6, 3, 6, 7, 19, 8, 9, 7, 20, 5, 21, 12, 6, 9, 5, 8, 22, 5, 4
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
EXAMPLE
The prime indices of 810 are {1,2,2,2,2,3}, with right half (inclusive) {2,2,3}, so a(810) = 7.
The prime indices of 3675 are {2,3,3,4,4}, with right half (inclusive) {3,4,4}, so a(3675) = 11.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Total[Take[prix[n], -Ceiling[Length[prix[n]]/2]]], {n, 100}]
CROSSREFS
Positions of first appearances are 1 and A001248. These partitions are counted by A360672 with rows reversed. First for prime indices, second for partitions, third for prime factors:
Product of the left half (exclusive) of the multiset of prime factors of n; a(1) = 0.
+10
17
0, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 2, 1, 4, 5, 2, 3, 2, 1, 2, 1, 4, 3, 2, 5, 4, 1, 2, 3, 4, 1, 2, 1, 2, 3, 2, 1, 4, 7, 2, 3, 2, 1, 6, 5, 4, 3, 2, 1, 4, 1, 2, 3, 8, 5, 2, 1, 2, 3, 2, 1, 4, 1, 2, 3, 2, 7, 2, 1, 4, 9, 2, 1, 4, 5, 2, 3
EXAMPLE
The prime factors of 250 are {2,5,5,5}, with left half (exclusive) {2,5}, with product 10, so a(250) = 10.
MATHEMATICA
Table[If[n==1, 0, Times@@Take[Join@@ConstantArray@@@FactorInteger[n], Floor[PrimeOmega[n]/2]]], {n, 100}]
a[n_] := Module[{p = Flatten[Table[#[[1]], {#[[2]]}] & /@ FactorInteger[n]]}, Times @@ p[[1 ;; Floor[Length[p]/2]]]]; a[1] = 0; Array[a, 100] (* Amiram Eldar, Nov 02 2024 *)
CROSSREFS
The right inclusive version A347044. A001221 counts distinct prime factors.
A006530 gives greatest prime factor.
First for prime indices, second for partitions, third for prime factors:
Product of the right half (exclusive) of the multiset of prime factors of n; a(1) = 0.
+10
17
0, 1, 1, 2, 1, 3, 1, 2, 3, 5, 1, 3, 1, 7, 5, 4, 1, 3, 1, 5, 7, 11, 1, 6, 5, 13, 3, 7, 1, 5, 1, 4, 11, 17, 7, 9, 1, 19, 13, 10, 1, 7, 1, 11, 5, 23, 1, 6, 7, 5, 17, 13, 1, 9, 11, 14, 19, 29, 1, 15, 1, 31, 7, 8, 13, 11, 1, 17, 23, 7, 1, 9, 1, 37, 5, 19, 11, 13, 1
EXAMPLE
The prime factors of 250 are {2,5,5,5}, with right half (exclusive) {5,5}, with product 25, so a(250) = 25.
MAPLE
f:= proc(n) local F;
F:= ifactors(n)[2];
F:= sort(map(t -> t[1]$t[2], F));
convert(F[ceil(nops(F)/2)+1 ..-1], `*`)
end proc:
f(1):= 0:
MATHEMATICA
Table[If[n==1, 0, Times@@Take[Join@@ConstantArray@@@FactorInteger[n], -Floor[PrimeOmega[n]/2]]], {n, 100}]
CROSSREFS
Positions of first appearances are A123666. The left inclusive version A347043. A001221 counts distinct prime factors.
A006530 gives greatest prime factor.
First for prime indices, second for partitions, third for prime factors:
Sum of the left half (exclusive) of the prime indices of n.
+10
16
0, 0, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 0, 2, 3, 1, 2, 1, 0, 1, 0, 2, 2, 1, 3, 2, 0, 1, 2, 2, 0, 1, 0, 1, 2, 1, 0, 2, 4, 1, 2, 1, 0, 3, 3, 2, 2, 1, 0, 2, 0, 1, 2, 3, 3, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 1, 4, 1, 0, 2, 4, 1, 0, 2, 3, 1, 2
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
EXAMPLE
The prime indices of 810 are {1,2,2,2,2,3}, with left half (exclusive) {1,2,2}, so a(810) = 5.
The prime indices of 3675 are {2,3,3,4,4}, with left half (exclusive) {2,3}, so a(3675) = 5.
MATHEMATICA
prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Table[Total[Take[prix[n], Floor[Length[prix[n]]/2]]], {n, 100}]
CROSSREFS
Positions of 0's are 1 and A000040. Positions of first appearances are 1 and A001248. First for prime indices, second for partitions, third for prime factors:
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