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Search: a331023 -id:a331023
Displaying 1-8 of 8 results found.
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A331024
Denominator: factorizations divided by strict factorizations A001055(n)/A045778(n).
+10
9
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 3, 1, 3, 1, 1, 1, 5, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 7, 1, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 9, 1, 1, 3, 4, 1, 1, 1, 3, 1, 1, 1, 9, 1, 1, 3, 3, 1, 1, 1, 7, 2, 1, 1, 9, 1, 1, 1, 5, 1, 9, 1, 3, 1, 1, 1, 10, 1, 3, 3, 5, 1, 1, 1, 5, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,8
COMMENTS
A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. It is strict if the factors are all different. Factorizations and strict factorizations are counted by A001055 and A045778 respectively.
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n
FORMULA
a(2^n) = A330995(n).
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[facs[n]]/Length[Select[facs[n], UnsameQ@@#&]], {n, 100}]//Denominator
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));
A045778(n, m=n) = ((n<=m) + sumdiv(n, d, if((d>1)&&(d<=m)&&(d<n), A045778(n/d, d-1))));
A331024(n) = denominator(A001055(n)/A045778(n)); \\ Antti Karttunen, May 27 2021
CROSSREFS
Positions of 1's include all elements of A001248 as well as A005117. The first position of a 1 that is not in A167207 is 128.
The numerators are A331023.
The rounded quotients are A331048.
The same for integer partitions is A330995.
Cf. A001055, A005117, A045778, A045779, A045780, A045782, A045783, A325755, A326028, A326622, A328966, A330972, A330977, A330991.
KEYWORD
nonn,frac
AUTHOR
Gus Wiseman, Jan 08 2020
EXTENSIONS
More terms from Antti Karttunen, May 27 2021
STATUS
approved
A331200
Least number with each record number of factorizations into distinct factors > 1.
+10
7
1, 6, 12, 24, 48, 60, 96, 120, 180, 240, 360, 480, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 4320, 5040, 7560, 8640, 10080, 15120, 20160, 25200, 30240, 40320, 45360, 50400, 55440, 60480, 75600, 90720, 100800, 110880, 120960, 151200, 181440, 221760
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
First differs from A330997 in lacking 64.
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..122
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.
EXAMPLE
Strict factorizations of the initial terms:
() (6) (12) (24) (48) (60) (96) (120)
(2*3) (2*6) (3*8) (6*8) (2*30) (2*48) (2*60)
(3*4) (4*6) (2*24) (3*20) (3*32) (3*40)
(2*12) (3*16) (4*15) (4*24) (4*30)
(2*3*4) (4*12) (5*12) (6*16) (5*24)
(2*3*8) (6*10) (8*12) (6*20)
(2*4*6) (2*5*6) (2*6*8) (8*15)
(3*4*5) (3*4*8) (10*12)
(2*3*10) (2*3*16) (3*5*8)
(2*4*12) (4*5*6)
(2*3*20)
(2*4*15)
(2*5*12)
(2*6*10)
(3*4*10)
(2*3*4*5)
MATHEMATICA
nn=1000;
strfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[strfacs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
qv=Table[Length[strfacs[n]], {n, nn}];
Table[Position[qv, i][[1, 1]], {i, Union[qv//.{foe___, x_, y_, afe___}/; x>y:>{foe, x, afe}]}]
CROSSREFS
A subset of A330997.
All terms belong to A025487.
This is the strict version of highly factorable numbers A033833.
The corresponding records are A331232(n) = A045778(a(n)).
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
The least number with A045779(n) strict factorizations is A045780(n)
Cf. A045783, A325238, A330972, A330973, A331023/A331024, A331201.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 12 2020
EXTENSIONS
a(37) and beyond from Giovanni Resta, Jan 17 2020
STATUS
approved
A330994
Numerator of P(n)/Q(n) = A000041(n)/A000009(n).
+10
6
1, 1, 2, 3, 5, 7, 11, 3, 11, 15, 21, 14, 77, 101, 135, 176, 231, 297, 385, 245, 627, 198, 1002, 1255, 1575, 979, 812, 1505, 1859, 4565, 1401, 3421, 2783, 1449, 6155, 4961, 17977, 21637, 26015, 31185, 1778, 2123, 26587, 63261, 75175, 44567, 17593, 8911, 49091
(list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
An integer partition of n is a finite, nonincreasing sequence of positive integers (parts) summing to n. It is strict if the parts are all different. Integer partitions and strict integer partitions are counted by A000041 and A000009 respectively.
LINKS
Table of n, a(n) for n=0..48.
FORMULA
A330994/A330995 = A000041/A000009.
MATHEMATICA
Table[PartitionsP[n]/PartitionsQ[n], {n, 0, 100}]//Numerator
CROSSREFS
The denominators are A330995.
The rounded quotients are A330996.
The same for factorizations is A331023.
Cf. A000009, A000041, A003238, A005117, A035359, A046063, A331022.
KEYWORD
nonn,frac
AUTHOR
Gus Wiseman, Jan 08 2020
STATUS
approved
A331232
Record numbers of factorizations into distinct factors > 1.
+10
6
1, 2, 3, 5, 7, 9, 10, 16, 18, 25, 34, 38, 57, 59, 67, 70, 91, 100, 117, 141, 161, 193, 253, 296, 306, 426, 552, 685, 692, 960, 1060, 1067, 1216, 1220, 1589, 1591, 1912, 2029, 2157, 2524, 2886, 3249, 3616, 3875, 4953, 5147, 5285, 5810, 6023, 6112, 6623, 8129
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..122
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.
FORMULA
a(n) = A045778(A331200(n)).
EXAMPLE
Representatives for the initial records and their strict factorizations:
() (6) (12) (24) (48) (60) (96) (120)
(2*3) (2*6) (3*8) (6*8) (2*30) (2*48) (2*60)
(3*4) (4*6) (2*24) (3*20) (3*32) (3*40)
(2*12) (3*16) (4*15) (4*24) (4*30)
(2*3*4) (4*12) (5*12) (6*16) (5*24)
(2*3*8) (6*10) (8*12) (6*20)
(2*4*6) (2*5*6) (2*6*8) (8*15)
(3*4*5) (3*4*8) (10*12)
(2*3*10) (2*3*16) (3*5*8)
(2*4*12) (4*5*6)
(2*3*20)
(2*4*15)
(2*5*12)
(2*6*10)
(3*4*10)
(2*3*4*5)
MATHEMATICA
nn=1000;
strfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[strfacs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
qv=Table[Length[strfacs[n]], {n, nn}];
Union[qv//.{foe___, x_, y_, afe___}/; x>y:>{foe, x, afe}]
PROG
(Python)
def fact(num):
....ret = []
....temp = num
....div = 2
....while temp > 1:
........while temp % div == 0:
............ret.append(div)
............temp //= div
........div += 1
....return ret
def all_partitions(lst):
....if lst:
........x = lst[0]
........for partition in all_partitions(lst[1:]):
............yield [x] + partition
............for i, _ in enumerate(partition):
................partition[i] *= x
................yield partition
................partition[i] //= x
....else:
........yield []
best = 0
terms = [0]
q = 2
while len(terms) < 100:
....total_set = set()
....factors = fact(q)
....total_set = set(tuple(sorted(x)) for x in all_partitions(factors) if len(x) == len(set(x)))
....if len(total_set) > best:
........best = len(total_set)
........terms.append(best)
........print(q, best)
....q += 2#only check evens
print(terms)
# David Consiglio, Jr., Jan 14 2020
CROSSREFS
The non-strict version is A272691.
The first appearance of a(n) in A045778 is at index A331200(n).
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
The least number with A045779(n) strict factorizations is A045780(n).
Cf. A033833, A045783, A325238, A330972, A330973, A330997, A331023/A331024, A331201.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 12 2020
EXTENSIONS
a(26)-a(37) from David Consiglio, Jr., Jan 14 2020
a(38) and beyond from Giovanni Resta, Jan 17 2020
STATUS
approved
A331201
Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.
+10
5
6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
First differs from A080257 in lacking 60.
LINKS
Table of n, a(n) for n=1..65.
EXAMPLE
Strict factorizations of selected terms:
(6) (12) (24) (48) (216)
(2*3) (2*6) (3*8) (6*8) (3*72)
(3*4) (4*6) (2*24) (4*54)
(2*12) (3*16) (6*36)
(2*3*4) (4*12) (8*27)
(2*3*8) (9*24)
(2*4*6) (12*18)
(2*108)
(3*8*9)
(4*6*9)
(2*3*36)
(2*4*27)
(2*6*18)
(2*9*12)
(3*4*18)
(3*6*12)
(2*3*4*9)
MATHEMATICA
strfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[strfacs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
Select[Range[100], PrimeQ[Length[strfacs[#]]]&]
CROSSREFS
The version for strict integer partitions is A035359.
The version for integer partitions is A046063.
The version for set partitions is A051130.
The non-strict version is A330991.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
Numbers whose number of strict factorizations is odd are A331230.
Numbers whose number of strict factorizations is even are A331231.
The least number with n strict factorizations is A330974(n).
Cf. A001318, A045780, A318286, A328966, A330992, A330993, A330997, A331023/A331024, A331050, A331051, A331200, A331232.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 12 2020
STATUS
approved
A331048
Nearest integer to A001055(n)/A045778(n), where A001055 is factorizations and A045778 is strict factorizations.
+10
4
1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,4
COMMENTS
A factorization of n is a finite, nondecreasing sequence of positive integers > 1 with product n. It is strict if the factors are all different.
LINKS
Table of n, a(n) for n=1..87.
MATHEMATICA
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Round[Length[facs[n]]/Length[Select[facs[n], UnsameQ@@#&]]], {n, 100}]
CROSSREFS
The exact quotient is A331023/A331024.
The same for integer partitions is A330996 ~ A330994/A330995.
Cf. A001055, A001222, A002033, A005117, A045778, A045779, A045780, A045782, A045783, A330972, A330977, A330991.
KEYWORD
nonn,frac
AUTHOR
Gus Wiseman, Jan 10 2020
STATUS
approved
A331231
Numbers k such that the number of factorizations of k into distinct factors > 1 is even.
+10
3
6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 144, 145, 146, 155, 158, 159, 160, 161, 166
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
First differs from A319238 in having 300.
LINKS
Table of n, a(n) for n=1..60.
MATHEMATICA
strfacs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[strfacs[n/d], Min@@#>d&]], {d, Rest[Divisors[n]]}]];
Select[Range[100], EvenQ[Length[strfacs[#]]]&]
CROSSREFS
The version for integer partitions is A001560.
The version for strict integer partitions is A090864.
The version for set partitions appears to be A016789.
The non-strict version is A331051.
The version for primes (instead of evens) is A331201.
The odd version is A331230.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
Cf. A001318, A045780, A045783, A318286, A328966, A330973, A330991, A330997, A331022, A331023/A331024, A331050, A331200, A331232.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 12 2020
STATUS
approved
A331198
Numbers n with exactly three times as many factorizations (A001055) as strict factorizations (A045778).
+10
0
128, 2187, 10368, 34992, 78125, 80000, 307328, 823543, 1250000, 1366875, 1874048, 3655808, 5250987, 6328125, 10690688, 13176688, 16681088, 19487171, 32019867, 35819648, 62462907, 62748517, 66706983, 90531968, 118210688, 182660427, 187578125, 239892608, 285012027
(list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Contains p^7 for all primes p.
LINKS
Table of n, a(n) for n=1..29.
EXAMPLE
The 15 factorizations and 5 strict factorizations of 2187:
(2187) (2187)
(27*81) (27*81)
(3*729) (3*729)
(9*243) (9*243)
(3*9*81) (3*9*81)
(9*9*27)
(3*27*27)
(3*3*243)
(3*9*9*9)
(3*3*3*81)
(3*3*9*27)
(3*3*3*9*9)
(3*3*3*3*27)
(3*3*3*3*3*9)
(3*3*3*3*3*3*3)
MATHEMATICA
facsm[n_]:=facsm[n]=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facsm[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Select[Range[100000], 3==Length[facsm[#]]/Length[Select[facsm[#], UnsameQ@@#&]]&]
CROSSREFS
Factorizations are A001055.
Strict factorizations are A045778.
Taking "twice" instead of "three times" gives A001248.
Cf. A045779, A045780, A045782, A045783, A330972, A331023/A331024, A331048, A331050.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 12 2020
EXTENSIONS
a(7)-(10) from Alois P. Heinz, Jan 17 2020
a(11)-a(29) from Giovanni Resta, Jan 20 2020
STATUS
approved
page 1

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