A006171 -id:A006171 - cp's OEIS frontend

A381455 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into a multiset of constant multisets.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 5, 5, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Mar 06 2025

nonn

First differs from A000688 at a(144) = 9, A000688(144) = 10.
First differs from A295879 at a(128) = 15, A295879(128) = 13.
Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into prime powers > 1.
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.
A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).
Multisets of constant multisets are generally not transitive. For example, we have arrows: {{1,1},{2}}: {1,1,2} -> {2,2} and {{2,2}}: {2,2} -> {4}, but there is no multiset of constant multisets {1,1,2} -> {4}.

			The prime indices of 36 are {1,1,2,2}, with the following 4 partitions into a multiset of constant multisets:
  {{1,1},{2,2}}
  {{1},{1},{2,2}}
  {{2},{2},{1,1}}
  {{1},{1},{2},{2}}
with block-sums: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, which are all different, so a(36) = 4.
The prime indices of 144 are {1,1,1,1,2,2}, with the following 10 partitions into a multiset of constant multisets:
  {{2,2},{1,1,1,1}}
  {{1},{2,2},{1,1,1}}
  {{2},{2},{1,1,1,1}}
  {{1,1},{1,1},{2,2}}
  {{1},{1},{1,1},{2,2}}
  {{1},{2},{2},{1,1,1}}
  {{2},{2},{1,1},{1,1}}
  {{1},{1},{1},{1},{2,2}}
  {{1},{1},{2},{2},{1,1}}
  {{1},{1},{1},{1},{2},{2}}
with block-sums: {4,4}, {1,3,4}, {2,2,4}, {2,2,4}, {1,1,2,4}, {1,2,2,3}, {2,2,2,2}, {1,1,1,1,4}, {1,1,2,2,2}, {1,1,1,1,2,2}, of which 9 are distinct, so a(144) = 9.
The a(n) partitions for n = 4, 8, 16, 32, 36, 64, 72, 128:
  (2)   (3)    (4)     (5)      (42)    (6)       (43)     (7)
  (11)  (21)   (22)    (32)     (222)   (33)      (322)    (43)
        (111)  (31)    (41)     (411)   (42)      (421)    (52)
               (211)   (221)    (2211)  (51)      (2221)   (61)
               (1111)  (311)            (222)     (4111)   (322)
                       (2111)           (321)     (22111)  (331)
                       (11111)          (411)              (421)
                                        (2211)             (511)
                                        (3111)             (2221)
                                        (21111)            (3211)
                                        (111111)           (4111)
                                                           (22111)
                                                           (31111)
                                                           (211111)
                                                           (1111111)

Robert Price, Table of n, a(n) for n = 1..1000

Before taking sums we had A000688.
Positions of 1 are A005117.
There is a chain from the prime indices of n to a singleton iff n belongs to A300273.
The lower version is A381453.
For distinct blocks we have A381715, before sum A050361.
For distinct block-sums we have A381716, before sums A381635 (zeros A381636).
Other multiset partitions of prime indices:
- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).
- For strict multiset partitions (A045778) see A381452.
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For set systems (A050326) see A381441 (upper).
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For set systems with distinct sums (A381633) see A381634, A293243.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
A000041 counts integer partitions, strict A000009.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002577, A002846, A018818, A213242, A213427, A275870, A289078, A299200, A299202, A300385.

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

a(s) = 1 for any squarefree number s.

a(p^k) = A000041(k) for any prime p.

A381871 Numbers whose prime indices cannot be partitioned into constant blocks having a common sum.

Original entry on oeis.org

6, 10, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 38, 39, 42, 44, 45, 46, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105, 106, 110

Offset: 1

Gus Wiseman, Mar 13 2025

nonn

First differs from A383100 in lacking 108.
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 numbers that cannot be written as a product of prime powers with equal sums of prime indices.
Partitions of this type are counted by A381993.

			The terms together with their prime indices begin:
    6: {1,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   26: {1,6}
   28: {1,1,4}
   30: {1,2,3}

Constant blocks: A000688, A006171, A279784, A295935, A381453 (lower), A381455 (upper).
Constant blocks with distinct sums: A381635, A381716.
For distinct instead of equal sums we have A381636, counted by A381717.
Partitions of this type are counted by A381993, complement A383093.
These are the positions of 0 in A381995.
A001055 counts multiset partitions of prime indices, strict A045778.
A050361 counts multiset partitions into distinct constant blocks.
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, A265947, A321469, A381633, A381715, A381719, A381806.

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[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Select[Range[100],Select[mps[prix[#]],SameQ@@Total/@#&&And@@SameQ@@@#&]=={}&]

A182269 Number of representations of n as a sum of products of pairs of positive integers, considered to be equivalent when terms or factors are reordered.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 14, 19, 31, 43, 65, 88, 132, 177, 253, 340, 478, 633, 874, 1150, 1562, 2045, 2736, 3553, 4713, 6082, 7969, 10234, 13301, 16973, 21889, 27789, 35577, 44961, 57179, 71906, 90950, 113874, 143204, 178592, 223505, 277599, 345822, 427934, 530797

Offset: 0

Alois P. Heinz, Apr 22 2012

nonn

From Yifan Xie, Sep 25 2024: (Start)
a(n) is the number of distinct sets A = {b_1, b_2, ..., b_n} such that 2*n positive integers x_1, x_2, ..., x_(2*n) exist where A = {x_1+x_2, x_3+x_4, ..., x_(2*n-1)+x_(2*n)} = {x_1*x_2, x_3*x_4, ..., x_(2*n-1)*x_(2*n)}.
Proof: Suppose that the number of sets A is b(n). Denote (x_(2*i-1)-1)*(x_(2*i)-1) by c_i. (1 <= i <= n)
Taking the sums of B and C, c_1 + c_2 + ... + c_n = n. (1)
Consider b_1, ..., b_n as n vertices, then the map B -> C is a directed graph G on these vertices, where each vertex has a source and a sink, so it can either be a cycle itself or decomposed into two or more cycles.
For the first case, the condition is equivalent to every proper subset of A' = {b_1, ..., b_n} is invalid for the corresponding n. Using (1), every partial sum of c_i is not equal to the number of addends. Therefore, c_i != 1. Then there must exist c_j = 0, hence c_i != 2. Then there must exist another c_k = 0, hence c_i != 3, and so on. Thus c_1, c_2, ..., c_n must be a permutation of 0, 0, ..., 0, n. Suppose that c_n = n, x_1 = x_3 = ... = x_(2*n-3) = 1. Since n has floor((A000005(n)+1)/2) ways to be expressed as the product of two positive integers, each product n = y*z means that x_(2*i-1) = y+1, x_(2*i) = z+1, thus there exists (y+1)*(z+1) in A, 1 + x_(2*l) = (a+1)*(b+1), 1*x_(2*l) = (a+1)*(b+1)-1, there exists (a+1)*(b+1)-1 in A, and so on until a+b+2 = (a+1)*(b+1)-1. In conclusion, there are floor((A000005(n)+1)/2) distinct A's in the second case, each of which is a group of consecutive integers. Denote the array by n = a*b .
For the second case, the array A can be decomposed into smaller arrays representing smaller n's, without breaking the structures of B and C. This process will finally end with all smaller arrays in the first case. Using the same notation, the arrays can be expressed as n = y_1*z_1 + y_2*z_2 + ... + y_s*z*s.
Therefore b(n) is the number of representations of n as a sum of products of pairs of unordered positive integers, hence b(n) = a(n). (End)

			a(0) = 1: 0 = the empty sum.
a(1) = 1: 1 = 1*1.
a(2) = 2: 2 = 1*1 + 1*1 = 1*2.
a(3) = 3: 3 = 1*1 + 1*1 + 1*1 = 1*1 + 1*2 = 1*3.
a(4) = 6: 4 = 1*1 + 1*1 + 1*1 + 1*1 = 1*1 + 1*1 + 1*2 = 1*1 + 1*3 = 1*2 + 1*2 = 2*2 = 1*4.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A000005, A006171, A038548, A066739, A182270, A211856, A211857.

with(numtheory):
a:= proc(n) option remember;  `if`(n=0, 1, add(add(
       d*ceil(tau(d)/2), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*Ceiling[DivisorSigma[0, d]/2], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 09 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[Product[Product[1/(1 - x^(k*j)), {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2019 *)

Euler transform of A038548.
G.f.: Product_{k>0} 1/(1-x^k)^A038548(k).
G.f.: Product_{k>=1} (Product_{j=1..k} 1/(1 - x^(k*j))). - Vaclav Kotesovec, Aug 19 2019

A211856 Number of representations of n as a sum of products of distinct pairs of positive integers, considered to be equivalent when terms or factors are reordered.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 6, 8, 10, 15, 20, 25, 34, 44, 56, 74, 94, 117, 151, 190, 236, 298, 370, 455, 567, 699, 853, 1050, 1282, 1555, 1898, 2299, 2770, 3351, 4035, 4837, 5811, 6952, 8288, 9898, 11782, 13978, 16600, 19660, 23225, 27451, 32366, 38074, 44799, 52609

Offset: 0

Alois P. Heinz, Apr 22 2012

nonn

			a(0) = 1: 0 = the empty sum.
a(1) = 1: 1 = 1*1.
a(2) = 1: 2 = 1*2.
a(3) = 2: 3 = 1*1 + 1*2 = 1*3.
a(4) = 3: 4 = 2*2 = 1*1 + 1*3 = 1*4.
a(5) = 4: 5 = 1*1 + 2*2 = 1*2 + 1*3 = 1*1 + 1*4 = 1*5.
a(6) = 6: 6 = 1*1 + 1*5 = 1*1 + 1*2 + 1*3 = 1*2 + 1*4 = 1*2 + 2*2 = 1*6 = 2*3
a(7) = 8: 7 = 1*1 + 1*2 + 1*4 = 1*1 + 1*2 + 2*2 = 1*1 + 1*6 = 1*1 + 2*3 = 1*2 + 1*5 = 1*3 + 1*4 = 1*3 + 2*2 = 1*7.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A000005, A006171, A038548, A066739, A182269, A182270, A211857.

with(numtheory):
b:= proc(n, i) option remember; local c;
      c:= ceil(tau(i)/2);
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)
       +add(b(n-i*j, i-1) *binomial(c, j), j=1..min(c, n/i))))
    end:
a:= n-> b(n, n):
seq(a(n), n=0..60);

b[n_, i_] := b[n, i] = Module[{c}, c = Ceiling[DivisorSigma[0, i]/2]; If[n == 0, 1, If[i < 1, 0, b[n, i-1] + Sum[b[n-i*j, i-1] *Binomial[c, j], {j, 1, Min[c, n/i]}]]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 09 2014, after Alois P. Heinz *)
nmax = 50; CoefficientList[Series[Product[Product[(1 + x^(k*j)), {j, 1, Min[k, nmax/k]}], {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2019 *)

G.f.: Product_{k>0} (1+x^k)^A038548(k). - Vaclav Kotesovec, Aug 19 2019

G.f.: Product_{k>=1} (Product_{j=1..k} (1 + x^(k*j))). - Vaclav Kotesovec, Aug 19 2019

A301554 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma_0(k)).

Original entry on oeis.org

1, 2, 6, 14, 32, 66, 138, 266, 512, 948, 1730, 3074, 5408, 9306, 15854, 26594, 44150, 72378, 117620, 189074, 301516, 476518, 747514, 1163470, 1798920, 2762040, 4215194, 6393196, 9642596, 14462518, 21581386, 32040562, 47345342, 69635866, 101974722, 148692638

Offset: 0

Vaclav Kotesovec, Mar 23 2018

nonn

Convolution of A006171 and A107742.

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

Cf. A000005, A006171, A107742, A320237.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((&*[(&*[(1 + x^(j*k))/(1-x^(j*k)): j in [1..(m+2)]]): k in [1..(m+2)]]))); // G. C. Greubel, Oct 29 2018

with(numtheory): seq(coeff(series(mul(((1+x^k)/(1-x^k))^sigma[0](k),k=1..n),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 29 2018

nmax = 50; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^DivisorSigma[0, k], {k, 1, nmax}], {x, 0, nmax}], x]

m=50; x='x+O('x^m); Vec(prod(k=1,m, prod(j=1,m+2, (1+x^(j*k))/(1-x^(j*k)) ))) \\ G. C. Greubel, Oct 29 2018

G.f.: Product_{i>=1, j>=1} (1 + x^(i*j))/(1 - x^(i*j)). - Ilya Gutkovskiy, May 23 2018

Conjecture: log(a(n)) ~ Pi * sqrt(n*log(n)/2). - Vaclav Kotesovec, Sep 03 2018

A381715 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into distinct constant blocks.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 10 2025

nonn

First differs from A050361 at a(1728) = 7, A050361(1728) = 8.

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.

			The prime indices of 1728 are {1,1,1,1,1,1,2,2,2}, with multiset partitions into distinct constant blocks:
  {{2,2,2},{1,1,1,1,1,1}}
  {{1},{2,2,2},{1,1,1,1,1}}
  {{2},{2,2},{1,1,1,1,1,1}}
  {{1,1},{2,2,2},{1,1,1,1}}
  {{1},{2},{2,2},{1,1,1,1,1}}
  {{1},{1,1},{1,1,1},{2,2,2}}
  {{2},{1,1},{2,2},{1,1,1,1}}
  {{1},{2},{1,1},{2,2},{1,1,1}}
with sums:
  {6,6}
  {1,5,6}
  {2,4,6}
  {2,4,6}
  {1,2,4,5}
  {1,2,3,6}
  {2,2,4,4}
  {1,2,2,3,4}
of which 7 are distinct, so a(1728) = 7.

Without distinct blocks (A000688) we have A381455, lower (A355731) A381453.
More on multiset partitions into constant blocks: A006171, A279784, A295935.
Positions of terms > 1 are A046099.
Before taking sums we had A050361.
For equal instead of distinct blocks we have A362421.
For strict instead of constant blocks we have A381441, before sums A050326.
For just distinct blocks we have A381452, before sums A045778.
For distinct sums we have A381716, before sums A381635, zeros A381636.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A002846, A005117, A050342, A213242, A213385, A293511, A299202, A300385, A317142, A381870.

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[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@#&&And@@SameQ@@@#&]]],{n,100}]

A004101 Number of partitions of n of the form a_1b_1^2 + a_2b_2^2 + ...; number of semisimple rings with p^n elements for any prime p.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 13, 18, 29, 40, 58, 79, 115, 154, 213, 284, 391, 514, 690, 900, 1197, 1549, 2025, 2600, 3377, 4306, 5523, 7000, 8922, 11235, 14196, 17777, 22336, 27825, 34720, 43037, 53446, 65942, 81423, 100033, 122991, 150481, 184149, 224449, 273614, 332291

Offset: 0

N. J. A. Sloane

The number of semisimple rings with p^n elements does not depend on the prime number p. - Paul Laubie, Mar 05 2024

			4 = 4*1^2 = 1*2^2 = 3*1^2 + 1*1^2 = 2*1^2 + 2*1^2 = 2*1^2 + 1*1^2 + 1*1^2 = 1*1^2 + 1*1^2 + 1*1^2 + 1*1^2.

J. Knopfmacher, Abstract Analytic Number Theory. North-Holland, Amsterdam, 1975, p. 293.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)
Gert Almkvist, Asymptotics of various partitions, arXiv:math/0612446 [math.NT], 2006 (section 6).
I. G. Connell, A number theory problem concerning finite groups and rings, Canad. Math. Bull, 7 (1964), 23-34.
I. G. Connell, Letter to N. J. A. Sloane, no date
N. J. A. Sloane, Transforms

Cf. A006171, A038538, A280451, A280661, A280662.

with(numtheory):
a:= proc(n) option remember;
      `if`(n=0, 1, add(add(d* mul(1+iquo(i[2], 2),
      i=ifactors(d)[2]), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Nov 26 2013
sqd:=proc(n) local t1,d; t1:=0; for d in divisors(n) do if (n mod d^2) = 0 then t1:=t1+1; fi; od; t1; end; # A046951
t2:=mul( 1/(1-x^n)^sqd(n),n=1..65); series(t2,x,60); seriestolist(%); # N. J. A. Sloane, Jun 24 2015

max = 45; A046951 = Table[Sum[Floor[n/k^2], {k, n}], {n, 0, max}] // Differences; f = Product[1/(1-x^n)^A046951[[n]], {n, 1, max}]; CoefficientList[Series[f, {x, 0, max}], x] (* Jean-François Alcover, Feb 11 2014 *)
nmax = 50; CoefficientList[Series[Product[1/(1 - x^(j*k^2)), {k, 1, Floor[Sqrt[nmax]] + 1}, {j, 1, Floor[nmax/k^2] + 1}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 03 2017 *)

N=66; x='x+O('x^N); gf=1/prod(j=1,N, eta(x^(j^2))); Vec(gf) /* Joerg Arndt, May 03 2008 */

EULER transform of A046951.

a(n) ~ exp(Pi^2 * sqrt(n) / 3 + sqrt(3/(2*Pi)) * Zeta(1/2) * Zeta(3/2) * n^(1/4) - 9 * Zeta(1/2)^2 * Zeta(3/2)^2 / (16*Pi^3)) * Pi^(3/4) / (sqrt(2) * 3^(1/4) * n^(5/8)) [Almkvist, 2006]. - Vaclav Kotesovec, Jan 03 2017

More terms, formula and better description from Christian G. Bower, Nov 15 1999

Name clarified by Paul Laubie, Mar 05 2024

A382076 Number of integer partitions of n whose run-sums are not all equal.

Original entry on oeis.org

0, 0, 0, 1, 1, 5, 6, 13, 15, 27, 37, 54, 64, 99, 130, 172, 220, 295, 372, 488, 615, 788, 997, 1253, 1547, 1955, 2431, 3005, 3706, 4563, 5586, 6840, 8332, 10139, 12305, 14879, 17933, 21635, 26010, 31181, 37314, 44581, 53156, 63259, 75163, 89124, 105553, 124752, 147210

Offset: 0

Gus Wiseman, Apr 02 2025

nonn

Also the number of integer partitions of n that cannot be partitioned into distinct constant multisets with a common sum. Multiset partitions of this type are ranked by A005117 /\ A326534 /\ A355743, while twice-partitions are counted by A382524, strict case of A279789.

			The partition (3,2,1,1,1) has runs ((3),(2),(1,1,1)) with sums (3,2,3) so is counted under a(8).
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)  (2211)   (331)     (431)
                      (21111)  (421)     (521)
                               (511)     (611)
                               (2221)    (3221)
                               (3211)    (3311)
                               (4111)    (4211)
                               (22111)   (5111)
                               (31111)   (22211)
                               (211111)  (32111)
                                         (311111)
                                         (2111111)

The complement is counted by A304442, ranks A353833.
For distinct instead of equal block-sums we have A381717.
This is the strict case of A381993, see A381995, zeros A381871.
A050361 counts factorizations into distinct prime powers, see A381715.
A304405 counts partitions with weakly decreasing run-sums, ranks A357875.
A304406 counts partitions with weakly increasing run-sums, ranks A357861.
A304428 counts partitions with strictly decreasing run-sums, ranks A357862.
A304430 counts partitions with strictly increasing run-sums, ranks A357864.
A317141 counts coarsenings of prime indices, refinements A300383.
A326534 ranks multiset partitions with a common sum.
A353837 counts partitions with distinct run-sums.
A354584 lists run-sums of weakly increasing prime indices.
A355743 ranks multiset partitions into constant blocks.
Cf. A000688, A005117, A006171, A047966, A279784, A381453, A381455, A381635, A381636, A381994, A382204.

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

More terms from Bert Dobbelaere, Apr 26 2025

A061257 Euler transform of reduced totient function psi(n), cf. A002322.

Original entry on oeis.org

1, 1, 2, 4, 7, 13, 21, 37, 58, 96, 153, 243, 376, 584, 897, 1353, 2046, 3060, 4552, 6714, 9862, 14386, 20898, 30198, 43427, 62159, 88600, 125804, 177881, 250615, 351819, 492203, 686294, 953954, 1321902, 1826394, 2516364, 3457332, 4737576, 6475332

Offset: 0

Vladeta Jovovic, Apr 21 2001

T. D. Noe, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Transforms

Cf. A002322, A061258, A006171, A001970, A061255, A061256.

nn = 20; b = Table[CarmichaelLambda[n], {n, nn}]; CoefficientList[Series[Product[1/(1 - x^m)^b[[m]], {m, nn}], {x, 0, nn}], x] (* T. D. Noe, Jun 19 2012 *)

G.f.: Product_{k=1..infinity} (1 - x^k)^(-psi(k)). a(n)=1/n*Sum_{k=1..n} a(n-k)*b(k), n>1, a(0)=1, b(k)=Sum_{d|k} d*psi(d), cf. A061258.

A182270 Number of representations of n as a sum of products of pairs of integers larger than 1, considered to be equivalent when terms or factors are reordered.

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 1, 0, 2, 1, 2, 0, 5, 1, 4, 2, 9, 2, 11, 3, 16, 7, 19, 6, 34, 13, 35, 18, 57, 23, 73, 32, 99, 53, 125, 60, 186, 92, 215, 127, 311, 164, 394, 221, 518, 320, 656, 386, 903, 545, 1091, 719, 1470, 925, 1863, 1215, 2390, 1642, 3015, 2037, 3966

Offset: 0

Alois P. Heinz, Apr 22 2012

nonn

			a(0) = 1: 0 = the empty sum.
a(1) = a(2) = a(3) = 0: no product is < 4.
a(4) = 1: 4 = 2*2.
a(6) = 1: 6 = 2*3.
a(8) = 2: 8 = 2*2 + 2*2 = 2*4.
a(9) = 1: 9 = 3*3.
a(12) = 5: 12 = 2*2 + 2*2 + 2*2 = 2*2 + 2*4 = 2*3 + 2*3 = 2*6 = 3*4.
a(13) = 1: 13 = 2*2 + 3*3.
a(14) = 4: 14 = 2*2 + 2*2 + 2*3 = 2*3 + 2*4 = 2*2 + 2*5 = 2*7.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A000005, A006171, A038548, A066739, A182269, A211856, A211857.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(
       d*(ceil(tau(d)/2)-1), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..70);

a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*(Ceiling[DivisorSigma[0, d]/2] - 1), {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 70}] (* Jean-François Alcover, Sep 09 2014, after Alois P. Heinz *)

Euler transform of A038548-1.
G.f.: Product_{k>0} 1/(1-x^k)^(A038548(k)-1).
G.f.: Product_{i>=1} Product_{j=2..i} 1/(1 - x^(i*j)). - Ilya Gutkovskiy, Sep 23 2019

cp's OEIS Frontend

A381455 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into a multiset of constant multisets.

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A381871 Numbers whose prime indices cannot be partitioned into constant blocks having a common sum.

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A182269 Number of representations of n as a sum of products of pairs of positive integers, considered to be equivalent when terms or factors are reordered.

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A211856 Number of representations of n as a sum of products of distinct pairs of positive integers, considered to be equivalent when terms or factors are reordered.

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A301554 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(sigma_0(k)).

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A381715 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into distinct constant blocks.

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A004101 Number of partitions of n of the form a_1*b_1^2 + a_2*b_2^2 + ...; number of semisimple rings with p^n elements for any prime p.

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A004101 Number of partitions of n of the form a_1b_1^2 + a_2b_2^2 + ...; number of semisimple rings with p^n elements for any prime p.