A006171 -id:A006171 - cp's OEIS frontend

A381995 Number of ways to partition the prime indices of n into constant blocks with a common sum.

1, 1, 1, 2, 1, 0, 1, 2, 2, 0, 1, 1, 1, 0, 0, 3, 1, 0, 1, 0, 0, 0, 1, 0, 2, 0, 2, 0, 1, 0, 1, 2, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 4, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 3, 0, 1, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Mar 19 2025

nonn

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

			The prime indices of 144 are {1,1,1,1,2,2}, with the following 2 multiset partitions into constant blocks with a common sum:
  {{2,2},{1,1,1,1}}
  {{2},{2},{1,1},{1,1}}
so a(144) = 2.

For just constant blocks we have A000688.
Twice-partitions of this type are counted by A279789.
For just a common sum we have A321455.
For distinct instead of equal sums we have A381635.
Positions of 0 are A381871, counted by A381993.
MM-numbers of these multiset partitions are A382215.
A001055 counts factorizations, strict A045778.
A050361 counts factorizations into distinct prime powers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
A353864 counts rucksack partitions, ranked by A353866.
Cf. A279784, A295935, A381453 (lower), A381455 (upper).
Cf. A000720, A000961, A001222, A006171, A265947, A381633, A381719.
Cf. A323774, A382076, A382204, A382524, A383014, A383093, A383309.

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@@SameQ@@@#&]],{n,100}]

A323774(n) = Sum_{A056239(k)=n} a(k). Gus Wiseman, Apr 25 2025

A383098 Number of integer partitions of n having at least one permutation with all equal run-sums.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 7, 2, 7, 5, 7, 2, 19, 2, 7, 8, 14, 2, 27, 2, 24, 8, 7, 2, 58, 5, 7, 13, 30, 2, 72, 2, 38, 8, 7, 8, 135, 2, 7, 8, 91, 2, 112, 2, 45, 38, 7, 2, 258, 5, 51, 8, 54, 2, 208, 8, 143, 8, 7, 2, 525, 2, 7, 44, 153, 8, 256, 2, 75, 8, 136, 2, 891, 2, 7, 57, 87, 8

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The partition (4,4,4,2,2,1,1,1,1) has permutations (4,2,2,4,1,1,1,1,4) and (4,1,1,1,1,4,2,2,4) so is counted under a(20).
The a(1) = 1 through a(10) = 7 partitions (A=10):
  1  2   3    4     5      6       7        8         9          A
     11  111  22    11111  33      1111111  44        333        55
              211          222              422       33111      22222
              1111         2211             2222      3111111    511111
                           3111             41111     111111111  2221111
                           21111            221111               22111111
                           111111           11111111             1111111111

For distinct instead of equal run-sums we appear to have A382427.
For run-lengths instead of sums we have A383013, ranked by complement of A382879.
The case of a unique choice is A383095, ranks A383099 = positions of 1 in A382877.
The complement is counted by A383096, ranks A383100 = positions of 0 in A382877.
These partitions are ranked by A383110.
The case of more than one choice is A383097, ranks A383015.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
Cf. A006171, A329738, A353832, A353839, A353850, A353932, A354584, A382076, A382857, A382876, A383094, A383112.

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

a(n) = A383097(n) + A383095(n), ranks A383015 \/ A383099.

More terms from Bert Dobbelaere, Apr 26 2025

A280540 G.f.: Product_{i>=1, j>=1} 1/(1 - x^(ij))^(ij).

Original entry on oeis.org

1, 1, 5, 11, 33, 67, 180, 366, 871, 1782, 3927, 7885, 16637, 32763, 66469, 128938, 253871, 484034, 930959, 1747304, 3292730, 6092664, 11282364, 20596790, 37568653, 67736175, 121886533, 217261372, 386216073, 681119439, 1197524035, 2091091902, 3639519280

Offset: 0

Vaclav Kotesovec, Jan 05 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. A000005, A006171, A038040, A061256, A107742, A192065, A280541.

nmax = 50; CoefficientList[Series[Product[1/(1-x^(i*j))^(i*j), {i, 1, nmax}, {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; s = 1 - x; Do[s *= Sum[Binomial[k*DivisorSigma[0, 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, Aug 27 2018 *)

G.f.: Product_{k>=1} 1/(1 - x^k)^(k*d(k)), where d(k) = number of divisors of k (A000005). - Ilya Gutkovskiy, Aug 26 2018

log(a(n)) ~ (3/2)^(2/3) * Zeta(3)^(1/3) * log(n)^(1/3) * n^(2/3). - Vaclav Kotesovec, Aug 28 2018

A280541 G.f.: Product_{i>=1, j>=1} (1 + x^(ij))^(ij).

Original entry on oeis.org

1, 1, 4, 10, 24, 52, 125, 253, 549, 1126, 2290, 4525, 8987, 17259, 33174, 62669, 117425, 217295, 399904, 726984, 1314257, 2354807, 4191671, 7405590, 13009916, 22696115, 39384232, 67937488, 116584833, 199001304, 338076500, 571507377, 961855945, 1611567819

Offset: 0

Vaclav Kotesovec, Jan 05 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. A000005, A006171, A038040, A061256, A107742, A192065, A280540.

nmax = 50; CoefficientList[Series[Product[(1+x^(i*j))^(i*j), {i, 1, nmax}, {j, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; s = 1 + x; Do[s *= Sum[Binomial[k*DivisorSigma[0, k], 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[s, x] (* Vaclav Kotesovec, Aug 27 2018 *)

G.f.: Product_{k>=1} (1 + x^k)^(k*d(k)), where d(k) = number of divisors of k (A000005). - Ilya Gutkovskiy, Aug 26 2018

Conjecture: log(a(n)) ~ 3 * Zeta(3)^(1/3) * log(n)^(1/3) * n^(2/3) / 2^(4/3). - Vaclav Kotesovec, Aug 29 2018

A383096 Number of integer partitions of n having no permutation with all equal run-sums.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 4, 13, 15, 25, 35, 54, 58, 99, 128, 168, 217, 295, 358, 488, 603, 784, 995, 1253, 1517, 1953, 2429, 2997, 3688, 4563, 5532, 6840, 8311, 10135, 12303, 14875, 17842, 21635, 26008, 31177, 37247, 44581, 53062, 63259, 75130, 89096, 105551, 124752, 147015, 173520

Offset: 0

Gus Wiseman, Apr 17 2025

nonn

			The a(3) = 1 through a(8) = 15 partitions:
  (21)  (31)  (32)    (42)   (43)      (53)
              (41)    (51)   (52)      (62)
              (221)   (321)  (61)      (71)
              (311)   (411)  (322)     (332)
              (2111)         (331)     (431)
                             (421)     (521)
                             (511)     (611)
                             (2221)    (3221)
                             (3211)    (3311)
                             (4111)    (4211)
                             (22111)   (5111)
                             (31111)   (22211)
                             (211111)  (32111)
                                       (311111)
                                       (2111111)

For distinct instead of equal run-sums we appear to have A381717, q.v.
For run-lengths instead of sums we have A382915, ranks A382879, by signature A382914.
For more than one permutation we have A383097, ranks A383015.
The complement is counted by A383098, ranks A383110
These partitions are ranked by A383100, positions of 0 in A382877.
Counting and ranking partitions by run-lengths and run-sums:
- constant: A047966 (ranks A072774), sums A304442 (ranks A353833)
- distinct: A098859 (ranks A130091), sums A353837 (ranks A353838)
- weakly decreasing: A100882 (ranks A242031), sums A304405 (ranks A357875)
- weakly increasing: A100883 (ranks A304678), sums A304406 (ranks A357861)
- strictly decreasing: A100881 (ranks A304686), sums A304428 (ranks A357862)
- strictly increasing: A100471 (ranks A334965), sums A304430 (ranks A357864)
A275870 counts collapsible partitions, ranks A300273.
A326534 ranks multiset partitions with a common sum, counted by A321455, normal A326518.
A353851 counts compositions with all equal run-sums, ranks A353848.
A382876 counts permutations of prime indices with distinct run-sums, zeros A381636.
A383095 counts partitions having a unique permutation with equal run-sums, ranks A383099.
Cf. A006171, A329738, A353832, A353839, A353932, A354584, A382076, A382857, A383094.

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

More terms from Bert Dobbelaere, Apr 26 2025

A280451 G.f.: Product_{k>=1, j>=1} (1+x^(j*k^2)).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 9, 13, 16, 20, 27, 34, 42, 53, 67, 82, 102, 125, 153, 188, 227, 274, 332, 401, 478, 574, 686, 815, 969, 1147, 1356, 1600, 1884, 2210, 2597, 3040, 3547, 4141, 4824, 5607, 6508, 7546, 8732, 10100, 11656, 13431, 15473, 17793, 20429, 23436

Offset: 0

Vaclav Kotesovec, Jan 03 2017

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A004101, A006171, A107742, A280663, A280664.

nmax = 100; CoefficientList[Series[Product[(1 + x^(j*k^2)), {k, 1, Floor[Sqrt[nmax]+1]}, {j, 1, Floor[nmax/k^2] + 1}], {x, 0, nmax}], x]

my(N=66, x='x+O('x^N)); Vec(prod(k=1, sqrt(N), eta(x^(2*k^2))/eta(x^(k^2)))) \\ Seiichi Manyama, Apr 29 2021

a(n) ~ exp(Pi^2*sqrt(n/2)/3 + sqrt(3) * (sqrt(2)-1) * Zeta(1/2) * Zeta(3/2) * n^(1/4) / (2^(3/4) * sqrt(Pi)) - 9*((sqrt(2)-1) * Zeta(1/2) * Zeta(3/2))^2 / (16*Pi^3)) * sqrt(Pi) / (2^(3/2) * sqrt(3) * n^(3/4)).

A293549 Expansion of Product_{k>=2} 1/(1 - x^k)^bigomega(k), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

Original entry on oeis.org

1, 0, 1, 1, 3, 2, 6, 5, 13, 12, 23, 24, 47, 47, 82, 92, 152, 167, 265, 301, 462, 532, 779, 914, 1324, 1548, 2174, 2590, 3573, 4250, 5771, 6904, 9254, 11092, 14638, 17606, 23043, 27680, 35820, 43155, 55383, 66642, 84850, 102141, 129171, 155394, 195134, 234679, 293184, 352096, 437359

Offset: 0

Ilya Gutkovskiy, Oct 11 2017

nonn

Euler transform of A001222.
Comment from R. J. Mathar, Sep 10 2021 (Start):
The triangle of the multiset transformation of A001222 looks as follows:
;1
0 ;0
1 0 ;1
1 0 0 ;1
2 1 0 0 ;3
1 1 0 0 0 ;2
2 3 1 0 0 0 ;6
1 3 1 0 0 0 0 ;5
3 6 3 1 0 0 0 0 ;13
2 5 4 1 0 0 0 0 0 ;12
2 9 8 3 1 0 0 0 0 0 ;23
1 9 9 4 1 0 0 0 0 0 0 ;24
3 14 17 9 3 1 0 0 0 0 0 0 ;47
1 12 18 11 4 1 0 0 0 0 0 0 0 ;47
2 17 29 21 9 3 1 0 0 0 0 0 0 0 ;82
...
The second column is A001222, the row sums (after the semicolons) are this sequence. (End)

N. J. A. Sloane, Transforms

Cf. A001222, A006171, A293548.

nmax = 50; CoefficientList[Series[Product[1/(1 - x^k)^PrimeOmega[k], {k, 2, nmax}], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d PrimeOmega[d], {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 50}]

G.f.: Product_{k>=2} 1/(1 - x^k)^b(k), where b(k) = [x^k] Sum_{p prime, j>=1} x^(p^j)/(1 - x^(p^j)).

a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} a(n-k)*b(k), b(k) = Sum_{d|k} d*bigomega(d).

A320778 Inverse Euler transform of the Euler totient function phi = A000010.

Original entry on oeis.org

1, 1, 0, 1, 0, 2, -3, 4, -4, 4, -9, 14, -19, 30, -42, 50, -76, 128, -194, 286, -412, 598, -909, 1386, -2100, 3178, -4763, 7122, -10758, 16414, -25061, 38056, -57643, 87568, -133436, 203618, -311128, 475536, -726355, 1109718, -1697766, 2601166, -3987903, 6114666

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, A320777, A320779, A320780, A320781, A320782.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-Totient(n))):
seq(a(n), n = 0..43); # 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[EulerPhi,30]]

A295739 Expansion of e.g.f. exp(Sum_{k>=1} d(k)*x^k/k!), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 1, 3, 9, 36, 158, 802, 4434, 26978, 176637, 1243528, 9316519, 74065506, 621187700, 5480130494, 50662481722, 489552042241, 4931215686119, 51668848043427, 561981734692781, 6333882472789914, 73850048237680936, 889461218944314524, 11051067390893340510

Offset: 0

Ilya Gutkovskiy, Nov 26 2017

nonn

Exponential transform of A000005.

Seiichi Manyama, Table of n, a(n) for n = 0..553
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, arXiv:math/0205301 [math.CO], 2002; Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Exponential Transform

Cf. A000005, A006171, A028342, A038200, A129921, A160399, A274804, A294363.

a:=series(exp(add(tau(k)*x^k/k!,k=1..100)),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 27 2019

nmax = 23; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k] x^k/k!, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] DivisorSigma[0, k] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]

E.g.f.: exp(Sum_{k>=1} A000005(k)*x^k/k!).

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1)*A000005(k)*a(n-k).

A301542 Expansion of Product_{k>=1} 1/(1 - x^k)^(sigma_4(k)).

Original entry on oeis.org

1, 1, 18, 100, 526, 2546, 12953, 60929, 282194, 1265959, 5580958, 24057117, 101922204, 424244720, 1739362261, 7027590168, 28017627428, 110295521903, 429110693519, 1650961520518, 6285554480496, 23693047787961, 88469251486817, 327380976530282, 1201122749057307

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..3315

Product_{k>=1} 1/(1 - x^k)^sigma_m(k): A006171 (m=0), A061256 (m=1), A275585 (m=2), A288391 (m=3), this sequence (m=4), A301543 (m=5), A301544 (m=6), A301545 (m=7), A301546 (m=8), A301547 (m=9).

Cf. A001159, A301548.

nmax = 40; CoefficientList[Series[Product[1/(1-x^k)^DivisorSigma[4, k], {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(2^(3/2) * 3^(2/3) * Pi * (Zeta(5)/7)^(1/6) * n^(5/6)/5 + Pi * (7/Zeta(5))^(1/6) * n^(1/6) / (240 * sqrt(2) * 3^(2/3)) - 3*Zeta(5) / (8*Pi^4)) * Zeta(5)^(1/12) / (2^(3/4) * 3^(2/3) * 7^(1/12) * n^(7/12)).

G.f.: exp(Sum_{k>=1} sigma_5(k)*x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Oct 26 2018

cp's OEIS Frontend

A381995 Number of ways to partition the prime indices of n into constant blocks with a common sum.

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A383098 Number of integer partitions of n having at least one permutation with all equal run-sums.

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A280540 G.f.: Product_{i>=1, j>=1} 1/(1 - x^(i*j))^(i*j).

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A280541 G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))^(i*j).

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A383096 Number of integer partitions of n having no permutation with all equal run-sums.

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A280451 G.f.: Product_{k>=1, j>=1} (1+x^(j*k^2)).

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A293549 Expansion of Product_{k>=2} 1/(1 - x^k)^bigomega(k), where bigomega(k) is the number of prime divisors of k counted with multiplicity (A001222).

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A320778 Inverse Euler transform of the Euler totient function phi = A000010.

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A295739 Expansion of e.g.f. exp(Sum_{k>=1} d(k)*x^k/k!), where d(k) is the number of divisors of k (A000005).

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A280540 G.f.: Product_{i>=1, j>=1} 1/(1 - x^(ij))^(ij).

A280541 G.f.: Product_{i>=1, j>=1} (1 + x^(ij))^(ij).