A006171 -id:A006171 - cp's OEIS frontend

A301543 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_5(k)).

1, 1, 34, 278, 1896, 13074, 92442, 607200, 3866890, 24062327, 146637082, 873517399, 5101981085, 29274370913, 165261721720, 918756928198, 5035250026792, 27229238821726, 145412875008092, 767414597651951, 4004930689994100, 20679955170511834, 105711772783426512

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..2331

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), this sequence (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A001160, A301549.

nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[5, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp((7*Pi)^(6/7) * (Zeta(7)/3)^(1/7) * n^(6/7) / (3*2^(3/7)) - Zeta'(-5)/2) * (Zeta(7)/(3*Pi))^(251/3528) / (2^(251/1176) * 7^(2015/3528) * n^(2015/3528)).

G.f.: exp(Sum_{k>=1} sigma_6(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

A301544 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_6(k)).

Original entry on oeis.org

1, 1, 66, 796, 7102, 70178, 702813, 6439533, 56938814, 495807251, 4218728690, 34991240657, 284295574638, 2269120791410, 17804772970005, 137455131596032, 1045354069608726, 7839809431539193, 58027706392726849, 424187792875896932, 3064539107659680502

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..1769

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), this sequence (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A013954, A301550.

nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[6, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(8 * 2^(3/8) * Pi * (Zeta(7)/15)^(1/8) * n^(7/8)/7 - Pi*(5/Zeta(7))^(1/8) * n^(1/8) / (504 * 2^(3/8) * 3^(7/8)) + 45*Zeta(7) / (16*Pi^6)) * Zeta(7)^(1/16) / (2^(29/16) * 15^(1/16) * n^(9/16)).

G.f.: exp(Sum_{k>=1} sigma_7(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

A301547 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_9(k)).

Original entry on oeis.org

1, 1, 514, 20198, 414696, 12465714, 373679122, 9181285000, 224372879810, 5583837482767, 132433701077938, 3028947042351535, 68425900639083569, 1518510622688185301, 32936878700790531296, 701684036762210944310, 14726705417058058788172, 304326729686784847885978

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..995

Cf. A006171 (m=0), A061256 (m=1), A275585 (m=2), A288391 (m=3), A301542 (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8).

Cf. A013957, A301553.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      sigma[9](d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Oct 26 2018

nmax = 30; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[9, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp((11*Pi)^(10/11) * (Zeta(11)/3)^(1/11) * n^(10/11) / (2^(3/11) * 5^(10/11)) - Zeta'(-9)/2) * (5*Zeta(11)/(3*Pi))^(131/2904) / (2^(131/968) * 11^(1583/2904) * n^(1583/2904)).

G.f.: exp(Sum_{k>=1} sigma_10(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

A382080 Number of ways to partition the prime indices of n into sets with a common sum.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 20 2025

nonn

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, sum A056239.

Also the number of factorizations of n into squarefree numbers > 1 with equal sums of prime indices.

			The prime indices of 900 are {1,1,2,2,3,3}, with the following partitions into sets with a common sum:
  {{1,2,3},{1,2,3}}
  {{3},{3},{1,2},{1,2}}
So a(900) = 2.

For just sets we have A050320, distinct A050326.
Twice-partitions of this type are counted by A279788.
For just a common sum we have A321455.
MM-numbers of these multiset partitions are A326534 /\ A302478.
For distinct instead of equal sums we have A381633.
For constant instead of strict blocks we have A381995.
Positions of 0 are A381719, counted by A381994.
A000688 counts factorizations into prime powers, distinct A050361.
A001055 counts factorizations, strict A045778.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000720, A000961, A001222, A006171, A279784, A353866, A381635, A381871.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Length[Select[mps[prix[n]], SameQ@@Total/@#&&And@@UnsameQ@@@#&]],{n,100}]

A323774 Number of multiset partitions, whose parts are constant and all have the same sum, of integer partitions of n.

Original entry on oeis.org

1, 1, 3, 3, 7, 3, 12, 3, 16, 8, 14, 3, 39, 3, 16, 15, 40, 3, 50, 3, 54, 17, 20, 3, 135, 10, 22, 25, 73, 3, 129, 3, 119, 21, 26, 19, 273, 3, 28, 23, 217, 3, 203, 3, 123, 74, 32, 3, 590, 12, 106, 27, 154, 3, 370, 23, 343, 29, 38, 3, 963, 3, 40, 95, 450, 25, 467, 3

Offset: 0

Gus Wiseman, Jan 27 2019

nonn

An unlabeled version of A279789.

			The a(1) = 1 through a(6) = 12 multiset partitions:
  (1)  (2)     (3)        (4)           (5)              (6)
       (11)    (111)      (22)          (11111)          (33)
       (1)(1)  (1)(1)(1)  (1111)        (1)(1)(1)(1)(1)  (222)
                          (2)(2)                         (3)(3)
                          (2)(11)                        (111111)
                          (11)(11)                       (3)(111)
                          (1)(1)(1)(1)                   (2)(2)(2)
                                                         (111)(111)
                                                         (2)(2)(11)
                                                         (2)(11)(11)
                                                         (11)(11)(11)
                                                         (1)(1)(1)(1)(1)(1)

Antti Karttunen, Table of n, a(n) for n = 0..20000

Cf. A001970, A006171 (constant parts), A007716, A034729, A047966 (uniform partitions), A047968, A279787, A279789 (twice-partitions version), A305551 (equal part-sums), A306017, A319056, A323766, A323775, A323776.

Table[Length[Join@@Table[Union[Sort/@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@ptn]],{ptn,Select[IntegerPartitions[n],SameQ@@#&]}]],{n,30}]

a(n) = if (n==0, 1, sumdiv(n, d, binomial(numdiv(d) + n/d - 1, n/d))); \\ Michel Marcus, Jan 28 2019

a(0) = 1; a(n) = Sum_{d|n} binomial(tau(d) + n/d - 1, n/d), where tau = A000005.

A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Gus Wiseman, Mar 22 2025

nonn

Also numbers with a unique factorization into prime powers with distinct sums of prime indices.

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, sum A056239.

			The prime indices of 270 are {1,2,2,2,3}, and there are two multiset partitions into constant multisets with distinct sums: {{1},{2},{3},{2,2}} and {{1},{3},{2,2,2}}, so 270 is not in the sequence.
The prime indices of 300 are {1,1,2,3,3}, of which there are no multiset partitions into constant multisets with distinct sums, so 300 is not in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, of which there is only one multiset partition into constant multisets with distinct sums: {{1},{1,1},{3},{2,2}}, so 360 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}

For distinct blocks instead of block-sums we have A004709, counted by A000726.
Twice-partitions of this type are counted by A279786.
MM-numbers of these multiset partitions are A326535 /\ A355743.
These are the positions of 1 in A381635.
For no choices we have A381636 (zeros of A381635), counted by A381717.
For strict instead of constant blocks we have A381870, counted by A382079.
Partitions of this type (unique into constant with distinct) are counted by A382301.
Normal multiset partitions of this type are counted by A382203.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower), A265947.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000688, A000720, A001222, A003963, A005117, A047966, A050361, A293511, A300385, A381716.
Cf. A006171, A279784, A295935.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Select[Range[100],Length[Select[pfacs[#],UnsameQ@@hwt/@#&]]==1&]

A383092 Number of integer partitions of n having at most one permutation with all equal run-lengths.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 7, 10, 13, 16, 22, 28, 34, 46, 58, 69, 90, 114, 141, 178, 216, 271, 338, 418, 506, 630, 769, 941, 1140, 1399, 1675, 2051, 2454, 2975, 3561, 4289, 5094, 6137, 7274, 8692, 10269, 12249, 14414, 17128, 20110, 23767, 27872, 32849, 38346, 45094, 52552, 61533

Offset: 0

Gus Wiseman, Apr 19 2025

nonn

			The partition (222211) has 1 permutation with all equal run-lengths: (221122), so is counted under a(10).
The partition (33211111) has no permutation with all equal run-lengths, so is counted under a(13).
The a(1) = 1 through a(7) = 10 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)
       (11)  (111)  (22)    (221)    (33)      (322)
                    (211)   (311)    (222)     (331)
                    (1111)  (2111)   (411)     (511)
                            (11111)  (3111)    (2221)
                                     (21111)   (4111)
                                     (111111)  (22111)
                                               (31111)
                                               (211111)
                                               (1111111)

For no choices we have A382915, ranks A382879.
For at least one choice we have A383013, for run-sums A383098, ranks A383110.
The complement is A383090, ranks A383089.
Partitions of this type are ranked by A383091 = positions of terms <= 1 in A382857.
For a unique choice we have A383094, ranks A383112.
For run-sums instead of lengths we have A383095 + A383096, ranks A383099 \/ A383100.
The complement for run-sums is A383097, ranks A383015, positions of terms > 1 in A382877.
Cf. A047966, A072774, A098859, A100471, A100881, A100882, A100883, A304442.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A239455 counts Look-and-Say partitions, ranks A351294, conjugate A381432.
A329738 counts compositions with equal run-lengths, ranks A353744.
A329739 counts compositions with distinct run-lengths, ranks A351596, complement A351291.
A351293 counts non-Look-and-Say partitions, ranks A351295, conjugate A381433.
Cf. A006171, A047993, A362608, A381871, A382076, A382771, A383111.

Table[Length[Select[IntegerPartitions[n],Length[Select[Permutations[#],SameQ@@Length/@Split[#]&]]<=1&]],{n,0,15}]

a(n) = A382915(n) + A383094(n).

More terms from Bert Dobbelaere, Apr 26 2025

A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(ijk*l)).

Original entry on oeis.org

1, 1, 5, 9, 29, 49, 135, 235, 565, 995, 2177, 3821, 7900, 13728, 26974, 46606, 88128, 150644, 276283, 467647, 835708, 1400874, 2448818, 4065230, 6975307, 11470265, 19359345, 31552473, 52488142, 84808548, 139274675, 223191639, 362297234, 576064732, 925295844

Offset: 0

Vaclav Kotesovec, Jan 04 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Lida Ahmadi, Ricardo Gómez Aíza, and Mark Daniel Ward, A unified treatment of families of partition functions, La Matematica (2024). Preprint available as arXiv:2303.02240 [math.CO], 2023.

Cf. A000041, A006171, A007426, A174465, A280486.

nmax=50; CoefficientList[Series[Product[1/(1-x^(i*j*k*l)), {i, 1, nmax}, {j, 1, nmax/i}, {k, 1, nmax/i/j}, {l, 1, nmax/i/j/k}], {x, 0, nmax}], x]
nmax = 50; tau4 = Table[DivisorSum[n, DivisorSigma[0, n/#] * DivisorSigma[0, #] &], {n, 1, nmax}]; s = 1 - x; Do[s *= Sum[Binomial[tau4[[k]], j]*(-1)^j*x^(j*k), {j, 0, nmax/k}]; s = Expand[s]; s = Take[s, Min[nmax + 1, Exponent[s, x] + 1, Length[s]]];, {k, 2, nmax}]; CoefficientList[Series[1/s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2018 *)

G.f.: Product_{k>=1} 1/(1 - x^k)^tau_4(k), where tau_4() = A007426. - Ilya Gutkovskiy, May 22 2018

A320776 Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, -1, -1, 0, 1, 0, -1, -1, -1, 1, 3, 3, -2, -5, -4, 0, 7, 7, 0, -9, -10, 2, 15, 15, -3, -27, -30, 3, 46, 51, 1, -71, -91, -7, 117, 157, 23, -194, -265, -57, 318, 465, 111, -536, -821, -230, 893, 1456, 505, -1485, -2559, -1036, 2433, 4483, 2022

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320777, A320778, A320779, A320780, A320781, A320782.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-NumberOfPrimeFactors(n))):
seq(a(n), n = 0..59); # Peter Luschny, Nov 21 2022

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[PrimeOmega,100]]

A320777 Inverse Euler transform of the number of distinct prime factors (without multiplicity) function A001221.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 0, 0, -1, -1, 1, 1, 0, -1, 0, 1, -1, -2, 1, 3, 1, -2, -2, 1, 0, -4, 0, 6, 6, -4, -8, 1, 4, -4, -5, 10, 16, -4, -25, -7, 17, 5, -16, 2, 42, 12, -58, -48, 40, 59, -27, -44, 67, 86, -103, -187, 36, 236, 45, -213, -5, 284, -23, -526, -188, 663, 520

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320778, A320779, A320780, A320781, A320782.

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[PrimeNu,100]]

cp's OEIS Frontend

A301543 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_5(k)).

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A301544 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_6(k)).

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A301547 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_9(k)).

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A382080 Number of ways to partition the prime indices of n into sets with a common sum.

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A323774 Number of multiset partitions, whose parts are constant and all have the same sum, of integer partitions of n.

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A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

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A383092 Number of integer partitions of n having at most one permutation with all equal run-lengths.

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A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(i*j*k*l)).

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A320776 Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.

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A280487 G.f.: Product_{i>=1, j>=1, k>=1, l>=1} 1/(1 - x^(ijk*l)).