A295935 -id:A295935 - cp's OEIS frontend

A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

1, 1, 3, 6, 14, 27, 58, 111, 223, 424, 817, 1527, 2870, 5279, 9710, 17622, 31877, 57100, 101887, 180406, 318106, 557453, 972796, 1688797, 2920123, 5026410, 8619551, 14722230, 25057499, 42494975, 71832114, 121024876, 203286806, 340435588, 568496753, 946695386

Offset: 0

N. J. A. Sloane

a(n) = number of partitions of n, when for each k there are p(k) different copies of part k. E.g., let the parts be 1, 2a, 2b, 3a, 3b, 3c, 4a, 4b, 4c, 4d, 4e, ... Then the a(4) = 14 partitions of 4 are: 4 = 4a = 4b = ... = 4e = 3a+1 = 3b+1 = 3c+1 = 2a+2a = 2a+2b = 2b+2b = 2a+1 = 2b+1 = 1+1+1+1.
Equivalently (Cayley), a(n) = number of 2-dimensional partitions of n. E.g., for n = 4 we have:
  4   31   3   22   2   211   21   2    2   1111   111   11    11   1
           1        2         1    11   1          1     11    1    1
                                        1                      1    1
                                                                    1
Also total number of different species of singularity for conjugate functions with n letters (Sylvester).
According to [Belmans], this sequence gives "[t]he number of Segre symbols for the intersection of two quadrics in a fixed dimension". - Eric M. Schmidt, Sep 02 2017
From Gus Wiseman, Jul 30 2022: (Start)
Also the number of non-isomorphic multiset partitions of weight n with all constant blocks. The strict case is A089259. For example, non-isomorphic representatives of the a(1) = 1 through a(3) = 6 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1},{1}}  {{1},{1,1}}
         {{1},{2}}  {{1},{2,2}}
                    {{1},{1},{1}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}
A000688 counts factorizations into prime powers.
A007716 counts non-isomorphic multiset partitions by weight.
A279784 counts twice-partitions of type PPR, factorizations A295935.
Constant partitions are ranked by prime-powers: A000961, A023894, A054685, A246655, A355743.
(End)

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + ...
a(3) = 6 because we have (111) = (111) = (11)(1) = (1)(1)(1), (12) = (12) = (1)(2), (3) = (3).
The a(4)=14 multiset partitions whose total sum of parts is 4 are:
((4)),
((13)), ((1)(3)),
((22)), ((2)(2)),
((112)), ((1)(12)), ((2)(11)), ((1)(1)(2)),
((1111)), ((1)(111)), ((11)(11)), ((1)(1)(11)), ((1)(1)(1)(1)). - _Gus Wiseman_, Dec 19 2016

A. Cayley, Recherches sur les matrices dont les termes sont des fonctions linéaires d'une seule indéterminée, J. Reine angew. Math., 50 (1855), 313-317; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 219.
V. A. Liskovets, Counting rooted initially connected directed graphs. Vesci Akad. Nauk. BSSR, ser. fiz.-mat., No 5, 23-32 (1969), MR44 #3927.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. J. Sylvester, An Enumeration of the Contacts of Lines and Surfaces of the Second Order, Phil. Mag. 1 (1851), 119-140. Reprinted in Collected Papers, Vol. 1. See p. 239, where one finds a(n)-2, but with errors.
J. J. Sylvester, Note on the 'Enumeration of the Contacts of Lines and Surfaces of the Second Order', Phil. Mag., Vol. VII (1854), pp. 331-334. Reprinted in Collected Papers, Vol. 2, pp. 30-33.

Reinhard Zumkeller, Table of n, a(n) for n = 0..5000 (first 500 terms from T. D. Noe)
Pieter Belmans, Segre symbols, 2016.
Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See pp. 10, 16.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 148
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).
L. Kaylor and D. Offner, Counting matrices over a finite field with all eigenvalues in the field, Involve, a Journal of Mathematics, Vol. 7 (2014), No. 5, 627-645. [DOI]
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious pairs, 2014.
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious numbers, IJNT, to appear.
XiKun Li, JunLi Li, Bin Liu and CongFeng Qiao, The parametric symmetry and numbers of the entangled class of 2 × M × N system, Science China Physics, Mechanics & Astronomy, Volume 54, Number 8, 1471-1475, DOI: 10.1007/s11433-011-4395-9.
Jessie Pitsillides, Segre Characteristic Equivalence, arXiv:2506.12065 [math.GM], 2025.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
N. J. A. Sloane, Transforms.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, vol. 2, vol. 3, vol. 4.
Index entries for sequences related to rooted trees

Related to A001383 via generating function.
The multiplicative version (factorizations) is A050336.
The ordered version (sequences of partitions) is A055887.
Row-sums of A061260.
Main diagonal of A055885.
We have A271619(n) <= a(n) <= A063834(n).
Column k=3 of A290353.
The strict case is A316980.
Cf. A000041, A061259, A006171, A061255, A061256, A061257, A089292, A000219.
Cf. A089300.
Cf. A001055, A072233, A112798, A275024.

Following Vladeta Jovovic:
a001970 n = a001970_list !! (n-1)
a001970_list = 1 : f 1 [1] where
   f x ys = y : f (x + 1) (y : ys) where
            y = sum (zipWith (*) ys a061259_list) `div` x
-- Reinhard Zumkeller, Oct 31 2015

with(combstruct); SetSetSetU := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
# second Maple program:
with(numtheory): with(combinat):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 19 2016

m = 32; f[x_] = Product[1/(1-x^k)^PartitionsP[k], {k, 1, m}]; CoefficientList[ Series[f[x], {x, 0, m-1}], x] (* Jean-François Alcover, Jul 19 2011, after g.f. *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k + x * O(x^n)), n))}; /* Michael Somos, Dec 20 2016 */

from sympy.core.cache import cacheit
from sympy import npartitions, divisors
@cacheit
def a(n): return 1 if n == 0 else sum([sum([d*npartitions(d) for d in divisors(j)])*a(n - j) for j in range(1, n + 1)]) / n
[a(n) for n in range(51)]  # Indranil Ghosh, Aug 19 2017, after Maple code
# (Sage) # uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 1)
a = EulerTransform(EulerTransform(b))
print([a(n) for n in range(36)]) # Peter Luschny, Nov 17 2022

G.f.: Product_{k >= 1} 1/(1-x^k)^p(k), where p(k) = number of partitions of k = A000041. [Cayley]
a(n) = (1/n)*Sum_{k = 1..n} a(n-k)*b(k), n > 1, a(0) = 1, b(k) = Sum_{d|k} d*numbpart(d), where numbpart(d) = number of partitions of d, cf. A061259. - Vladeta Jovovic, Apr 21 2001
Logarithmic derivative yields A061259 (equivalent to above formula from Vladeta Jovovic). - Paul D. Hanna, Sep 05 2012
a(n) = Sum_{k=1..A000041(n)} A001055(A215366(n,k)) = number of factorizations of Heinz numbers of integer partitions of n. - Gus Wiseman, Dec 19 2016
a(n) = |{m>=1 : n = Sum_{k=1..A001222(m)} A056239(A112798(m,k)+1)}| = number of normalized twice-prime-factored multiset partitions (see A275024) whose total sum of parts is n. - Gus Wiseman, Dec 19 2016

Additional comments from Valery A. Liskovets

Sylvester references from Barry Cipra, Oct 07 2003

A055887 Number of ordered partitions of partitions.

Original entry on oeis.org

1, 1, 3, 8, 22, 59, 160, 431, 1164, 3140, 8474, 22864, 61697, 166476, 449210, 1212113, 3270684, 8825376, 23813776, 64257396, 173387612, 467856828, 1262431711, 3406456212, 9191739970, 24802339472, 66924874539, 180585336876, 487278670744, 1314838220172

Offset: 0

Christian G. Bower, Jun 09 2000

nonn

Jordan matrices are upper bidiagonal matrices such that (A) the diagonal entries are in sorted order, (B) there are only 1's and 0's on the superdiagonal, (C) for each superdiagonal 1, the two diagonal entries to the left and below it must be equal. Let J(N) be the number of N X N Jordan matrices where the diagonal values are, without loss of generality, taken to be a prefix of some fixed strictly increasing sequence x_1, x_2, x_3, ... If Jordan blocks sorted by eigenvalue with ties broken by block size during the sorting, then J(1, 2, 3, ...) is this sequence. - Warren D. Smith, Jan 28 2002
Number of compositions of n into parts k >= 1 where there are A000041(k) sorts of part k. - Joerg Arndt, Sep 30 2012
Also number of chains of multisets that partition a normal multiset of weight n, where a multiset is normal if it spans an initial interval of positive integers. - Gus Wiseman, Oct 28 2015
From Gus Wiseman, Jul 31 2022: (Start)
Also the number of ways to choose a multiset partition into constant multisets of a multiset of length n covering an initial interval of positive integers. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(3) = 8 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1},{1}}  {{1},{1,1}}
         {{1},{2}}  {{1},{2,2}}
                    {{2},{1,1}}
                    {{1},{1},{1}}
                    {{1},{1},{2}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}
Factorizations into prime powers, are counted by A000688.
The strongly normal case is A063834.
The strongly normal strict case is A270995.
Twice-partitions of type PPR are counted by A279784, factorizations A295935.
The strict case is A304969.
(End)

			The a(4) = 22 chains of multisets, where notation x-y means "y is a submultiset of x", are: (o-o-o-o) (oo-o-o) (oo-oo) (ooo-o) (oooo) (oe-o-o) (ooe-o) (oooe) (oe-oe) (ooe-e) (oee-o) (ooee) (oei-o) (ooei) (oe-e-e) (oee-e) (oeee) (oei-e) (oeei) (oei-i) (oeii) (oeis).
From _Gus Wiseman_, Jul 31 2022: (Start)
a(n) is the number of ways to choose an integer partition of each part of an integer composition of n. The a(0) = 1 through a(3) = 8 choices are:
  ()  ((1))  ((2))     ((3))
             ((11))    ((21))
             ((1)(1))  ((111))
                       ((1)(2))
                       ((2)(1))
                       ((1)(11))
                       ((11)(1))
                       ((1)(1)(1))
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..2320
N. J. A. Sloane, Transforms
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89, DOI:10.1007/s11083-004-9460-9.
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol.

Cf. A055888, A083355, A095975, A246828.
Row sums of A060642.
Cf. A326346.
The unordered version is A001970, row-sums of A061260.
A000041 counts integer partitions, strict A000009.
A011782 counts integer compositions.
A072233 counts partitions by sum and length.
Cf. A001055, A050361, A063834, A279784, A355743.

with(combstruct); SeqSetSetU := [T, {T=Sequence(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
P := (x) -> product( 1/(1-x^k), k=1..20 ) - 1; F := (x) -> series( 1/(1-P(x)) - 1, x, 21 ); # F(x) is g.f. for this sequence # Warren D. Smith, Jan 28 2002
A055887rec:= proc(n::integer) local k; option remember; with(combinat): if n = 0 then 1 else add(numbpart(k) *procname(n - k), k=1..n); end if; end proc: seq (A055887rec(n), n=0..10); # Thomas Wieder, Nov 26 2007

a = 1/Product[(1 - x^k), {k, 1, \[Infinity]}] - 1; CoefficientList[Series[1/(1 - a), {x, 0, 20}], x] (* Geoffrey Critzer, Dec 23 2010 *)
(1/(2 - 1/QPochhammer[x]) + O[x]^30)[[3]] (* Vladimir Reshetnikov, Sep 22 2016 *)
Table[Sum[Times@@PartitionsP/@c,{c,Join@@Permutations/@IntegerPartitions[n]}],{n,0,10}] (* Gus Wiseman, Jul 31 2022 *)

Vec(1/(2-1/eta(x+O(x^66)))) \\ Joerg Arndt, Sep 30 2012

Invert transform of partitions numbers A000041.
Let p(k) be the number of integer partitions of k. Furthermore, set a(0)=1. Then a(n) = Sum_{k=1..n} p(k)*a(n-k). - Thomas Wieder, Nov 26 2007
G.f.: 1/( 1 - Sum_{k>=1} p(k)*x^k ) where p(k) = A000041(k) is the number of integer partitions of k. - Joerg Arndt, Sep 30 2012
a(n) ~ c * d^n, where d = 2.698329106474211231263998666188376330713465125913986356769... (see A246828) and c = 0.414113793172792357745578049739573823627306487211379286647... - Vaclav Kotesovec, Mar 29 2014
G.f.: 1/(2 - 1/(x)inf), where (x)_inf is the q-Pochhammer symbol. - _Vladimir Reshetnikov, Sep 22 2016

A050361 Number of factorizations into distinct prime powers greater than 1.

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

Offset: 1

Christian G. Bower, Oct 15 1999

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).

The number of unordered factorizations of n into 1 and exponentially odd prime powers, i.e., p^e where p is a prime and e is odd (A246551). - Amiram Eldar, Jun 12 2025

			From _Gus Wiseman_, Jul 30 2022: (Start)
The A000688(216) = 9 factorizations of 216 into prime powers are:
  (2*2*2*3*3*3)
  (2*2*2*3*9)
  (2*2*2*27)
  (2*3*3*3*4)
  (2*3*4*9)
  (2*4*27)
  (3*3*3*8)
  (3*8*9)
  (8*27)
Of these, the a(216) = 4 strict cases are:
  (2*3*4*9)
  (2*4*27)
  (3*8*9)
  (8*27)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..100000 (first 10000 terms from Reinhard Zumkeller)
Index entries for sequences computed from exponents in factorization of n.

Cf. A000009, A050360, A050362, A050363, A050364, A101296, A246551.
Cf. A124010.
This is the strict case of A000688.
Positions of 1's are A004709, complement A046099.
The case of primes (instead of prime-powers) is A008966, non-strict A000012.
The non-strict additive version allowing 1's A023893, ranked by A302492.
The non-strict additive version is A023894, ranked by A355743.
The additive version (partitions) is A054685, ranked by A356065.
The additive version allowing 1's is A106244, ranked by A302496.
A001222 counts prime-power divisors.
A005117 lists all squarefree numbers.
A034699 gives maximal prime-power divisor.
A246655 lists all prime-powers (A000961 includes 1), towers A164336.
A296131 counts twice-factorizations of type PQR, non-strict A295935.
Cf. A001970, A002110, A025487, A055887, A063834, A076610, A085970, A279786, A302590, A302601, A354911, A355742.

a050361 = product . map a000009 . a124010_row
-- Reinhard Zumkeller, Aug 28 2014

A050361 := proc(n)
    local a,f;
    if n = 1 then
        1;
    else
        a := 1 ;
        for f in ifactors(n)[2] do
            a := a*A000009(op(2,f)) ;
        end do:
    end if;
end proc: # R. J. Mathar, May 25 2017

Table[Times @@ PartitionsQ[Last /@ FactorInteger[n]], {n, 99}] (* Arkadiusz Wesolowski, Feb 27 2017 *)

A000009(n,k=(n-!(n%2))) = if(!n,1,my(s=0); while(k >= 1, if(k<=n, s += A000009(n-k,k)); k -= 2); (s));
A050361(n) = factorback(apply(A000009,factor(n)[,2])); \\ Antti Karttunen, Nov 17 2019

Dirichlet g.f.: Product_{n is a prime power >1}(1 + 1/n^s).
Multiplicative with a(p^e) = A000009(e).
a(A002110(k))=1.
a(n) = A050362(A101296(n)). - R. J. Mathar, May 26 2017
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.26020571070524171076..., where f(x) = (1-x) * Product_{k>=1} (1 + x^k). - Amiram Eldar, Oct 03 2023

A023893 Number of partitions of n into prime power parts (1 included); number of nonisomorphic Abelian subgroups of symmetric group S_n.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 36, 48, 63, 82, 105, 134, 171, 215, 269, 335, 415, 511, 626, 764, 929, 1125, 1356, 1631, 1953, 2333, 2776, 3296, 3903, 4608, 5427, 6377, 7476, 8744, 10205, 11886, 13818, 16032, 18565, 21463, 24768, 28536

Offset: 0

Olivier Gérard

nonn

			From _Gus Wiseman_, Jul 28 2022: (Start)
The a(0) = 1 through a(6) = 10 partitions:
  ()  (1)  (2)   (3)    (4)     (5)      (33)
           (11)  (21)   (22)    (32)     (42)
                 (111)  (31)    (41)     (51)
                        (211)   (221)    (222)
                        (1111)  (311)    (321)
                                (2111)   (411)
                                (11111)  (2211)
                                         (3111)
                                         (21111)
                                         (111111)
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for sequences related to groups

Cf. A009490, A023894 (first differences), A062297 (number of Abelian subgroups).
Cf. A018819, A062051, A131995.
The multiplicative version (factorizations) is A000688.
Not allowing 1's gives A023894, strict A054685, ranked by A355743.
The version for just primes (not prime-powers) is A034891, strict A036497.
The strict version is A106244.
These partitions are ranked by A302492.
A000041 counts partitions, strict A000009.
A001222 counts prime-power divisors.
A072233 counts partitions by sum and length.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
Cf. A001970, A055887, A063834, A085970, A279784, A295935, A356065.

Table[Length[Select[IntegerPartitions[n],Count[Map[Length,FactorInteger[#]], 1] == Length[#] &]], {n, 0, 35}] (* Geoffrey Critzer, Oct 25 2015 *)
nmax = 50; Clear[P]; P[m_] := P[m] = Product[Product[1/(1-x^(p^k)), {k, 1, m}], {p, Prime[Range[PrimePi[nmax]]]}]/(1-x)+O[x]^nmax // CoefficientList[ #, x]&; P[1]; P[m=2]; While[P[m] != P[m-1], m++]; P[m] (* Jean-François Alcover, Aug 31 2016 *)

lista(m) = {x = t + t*O(t^m); gf = prod(k=1, m, if (isprimepower(k), 1/(1-x^k), 1))/(1-x); for (n=0, m, print1(polcoeff(gf, n, t), ", "));} \\ Michel Marcus, Mar 09 2013

from functools import lru_cache
from sympy import factorint
@lru_cache(maxsize=None)
def A023893(n):
    @lru_cache(maxsize=None)
    def c(n): return sum((p**(e+1)-p)//(p-1) for p,e in factorint(n).items())+1
    return (c(n)+sum(c(k)*A023893(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024

G.f.: (Product_{p prime} Product_{k>=1} 1/(1-x^(p^k))) / (1-x).

A023894 Number of partitions of n into prime power parts (1 excluded).

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 15, 19, 23, 29, 37, 44, 54, 66, 80, 96, 115, 138, 165, 196, 231, 275, 322, 380, 443, 520, 607, 705, 819, 950, 1099, 1268, 1461, 1681, 1932, 2214, 2533, 2898, 3305, 3768, 4285, 4872, 5530, 6267, 7094, 8022, 9060

Offset: 0

Olivier Gérard

nonn

			From _Gus Wiseman_, Jul 28 2022: (Start)
The a(0) = 1 through a(9) = 7 partitions:
  ()  .  (2)  (3)  (4)   (5)   (33)   (7)    (8)     (9)
                   (22)  (32)  (42)   (43)   (44)    (54)
                               (222)  (52)   (53)    (72)
                                      (322)  (332)   (333)
                                             (422)   (432)
                                             (2222)  (522)
                                                     (3222)
(End)

Seiichi Manyama, Table of n, a(n) for n = 0..10000
E. Grosswald, Partitions into prime powers

The multiplicative version (factorizations) is A000688, coprime A354911.
Allowing 1's gives A023893, strict A106244, ranked by A302492.
The strict version is A054685.
The version for just primes is ranked by A076610, squarefree A356065.
Twice-partitions of this type are counted by A279784, factorizations A295935.
These partitions are ranked by A355743.
A000041 counts partitions, strict A000009.
A001222 counts prime-power divisors.
A072233 counts partitions by sum and length.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
Cf. A001970, A055887, A063834, A085970.

Table[Length[Select[IntegerPartitions[n],And@@PrimePowerQ/@#&]],{n,0,30}] (* Gus Wiseman, Jul 28 2022 *)

is_primepower(n)= {ispower(n, , &n); isprime(n)}
lista(m) = {x = t + t*O(t^m); gf = prod(k=1, m, if (is_primepower(k), 1/(1-x^k), 1)); for (n=0, m, print1(polcoeff(gf, n, t), ", "));}
\\ Michel Marcus, Mar 09 2013

from functools import lru_cache
from sympy import factorint
@lru_cache(maxsize=None)
def A023894(n):
    @lru_cache(maxsize=None)
    def c(n): return sum((p**(e+1)-p)//(p-1) for p,e in factorint(n).items())
    return (c(n)+sum(c(k)*A023894(n-k) for k in range(1,n)))//n if n else 1 # Chai Wah Wu, Jul 15 2024

G.f.: Prod(p prime, Prod(k >= 1, 1/(1-x^(p^k))))

A381636 Numbers whose prime indices cannot be partitioned into constant blocks with distinct sums.

Original entry on oeis.org

12, 60, 63, 84, 120, 126, 132, 156, 204, 228, 252, 276, 300, 315, 325, 348, 372, 420, 444, 492, 504, 516, 560, 564, 588, 630, 636, 650, 660, 693, 708, 720, 732, 780, 804, 819, 840, 852, 876, 924, 931, 948, 975, 996, 1008, 1020, 1068, 1071, 1092, 1140, 1164

Offset: 1

Gus Wiseman, Mar 10 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.
Also numbers that cannot be written as a product of prime powers > 1 with distinct sums of prime indices (A056239).
Contains no squarefree numbers.
Conjecture: These are the zeros of A382876.

			The prime indices of 300 are {1,1,2,3,3}, with partitions into constant blocks:
  {{2},{1,1},{3,3}}
  {{1},{1},{2},{3,3}}
  {{2},{3},{3},{1,1}}
  {{1},{1},{2},{3},{3}}
but none of these has distinct block-sums, so 300 is in the sequence.
The terms together with their prime indices begin:
   12: {1,1,2}
   60: {1,1,2,3}
   63: {2,2,4}
   84: {1,1,2,4}
  120: {1,1,1,2,3}
  126: {1,2,2,4}
  132: {1,1,2,5}
  156: {1,1,2,6}
  204: {1,1,2,7}
  228: {1,1,2,8}
  252: {1,1,2,2,4}
  276: {1,1,2,9}
  300: {1,1,2,3,3}

More on multiset partitions into constant blocks: A006171, A279784, A295935.
These are the positions of 0 in A381635, after taking block-sums A381716.
Partitions of this type are counted by A381717.
For strict instead of constant blocks we have A381806, zeros of A381633.
For equal instead of distinct block-sums we have A381871.
A000688 counts multiset partitions into constant, see A381455 (upper), A381453 (lower).
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A050361 counts multiset partitions into distinct constant blocks, after sums A381715.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A005117, A050320, A059404, A213242, A293243, A299202, A300385, A381078, A381454, A381634.

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],Select[pfacs[#],UnsameQ@@hwt/@#&]=={}&]

A381635 Number of ways to partition the prime indices of n into constant blocks with distinct sums.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 09 2025

nonn

First differs from A381716 at a(1728) = 5, A381716(1728) = 4.

Also the number of factorizations on n into prime powers > 1 with distinct sums of prime indices (A056239).

			The a(432) = 3 multiset partitions:
  {{2,2,2},{1,1,1,1}}
  {{1},{1,1,1},{2,2,2}}
  {{1},{2},{2,2},{1,1,1}}
Note {{2},{2,2},{1,1,1,1}} is not included, as it does not have distinct block-sums.

Without distinct block-sums we have A000688, after sums A381455 (upper), A381453 (lower).
For distinct blocks instead of sums we have A050361, after sums A381715.
For strict instead of constant we have A381633 (zeros A381806), after sums A381634.
Positions of 0 are A381636.
Taking block-sums (and sorting) gives A381716.
Other multiset partitions of prime indices:
More on multiset partitions into constant blocks: A006171, A279784, A295935.
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.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A002846, A005117, A213242, A213385, A293511, A299202, A300385, A317142, A381870.

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]}]];
Table[Length[Select[pfacs[n],UnsameQ@@hwt/@#&]],{n,100}]

A381716 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 constant blocks with distinct sums.

Original entry on oeis.org

Offset: 1

Gus Wiseman, Mar 10 2025

nonn

First differs from A381635 at a(1728) = 4, A381635(1728) = 5.

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 constant multisets with distinct sums:
  {{1,1,1,1,1,1},{2,2},{2}}
  {{1,1,1,1,1},{1},{2,2,2}}
  {{1,1,1,1,1},{1},{2,2},{2}}
  {{1,1,1,1},{1,1},{2,2,2}}
  {{1,1,1},{1,1},{1},{2,2,2}}
with block-sums: {1,5,6}, {2,4,6}, {1,2,3,6}, {1,2,4,5}, so a(1728) = 4.

Without distinct sums we have A000688, after sums A381455 (upper), A381453 (lower).
More on multiset partitions into constant blocks: A006171, A279784, A295935.
For strict instead of constant we have A381633, before sums A381634.
Before taking sums we had A381635.
Positions of 0 are A381636.
For distinct blocks instead of sums we have A381715.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower).
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000720, A001222, A002846, A005117, A050361, A213242, A265947, A293511, A299202, A300385, A362421.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn,{ptn,IntegerPartitions[Length[y]]}];
Table[Length[Union[Sort[Total/@#]&/@Select[Join@@@Tuples[mce/@Split[prix[n]]],UnsameQ@@Total/@#&]]],{n,100}]

A355743 Numbers whose prime indices are all prime-powers.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25, 27, 31, 33, 35, 41, 45, 49, 51, 53, 55, 57, 59, 63, 67, 69, 75, 77, 81, 83, 85, 93, 95, 97, 99, 103, 105, 109, 115, 119, 121, 123, 125, 127, 131, 133, 135, 147, 153, 155, 157, 159, 161, 165, 171, 175, 177, 179, 187

Offset: 1

Gus Wiseman, Jul 24 2022

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.

Also MM-numbers of multiset partitions into constant multisets, where the multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The terms together with their prime indices begin:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  31: {11}
  33: {2,5}
  35: {3,4}
  41: {13}
  45: {2,2,3}

Robert Price, Table of n, a(n) for n = 1..1410

The multiplicative version is A000688, strict A050361, coprime A354911.
The case of only primes (not all prime-powers) is A076610, strict A302590.
Allowing prime index 1 gives A302492.
These are the products of elements of A302493.
Requiring n to be a prime-power gives A302601.
These are the positions of 1's in A355741.
The squarefree case is A356065.
The complement is A356066.
A001222 counts prime-power divisors.
A023894 counts ptns into prime-powers, strict A054685, with 1's A023893.
A034699 gives maximal prime-power divisor.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
A355742 chooses a prime-power divisor of each prime index.
Cf. A085970, A106244, A279784, A295935, A355731, A356064.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@PrimePowerQ/@primeMS[#]&]

A381453 Number of multisets that can be obtained by choosing a constant integer partition of each prime index of n and taking the multiset union.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 3, 4, 1, 2, 3, 4, 2, 6, 2, 3, 2, 3, 4, 4, 3, 4, 4, 2, 1, 4, 2, 6, 3, 6, 4, 8, 2, 2, 6, 4, 2, 6, 3, 4, 2, 6, 3, 4, 4, 5, 4, 4, 3, 8, 4, 2, 4, 6, 2, 8, 1, 8, 4, 2, 2, 6, 6, 6, 3, 4, 6, 6, 4, 6, 8, 4, 2, 5, 2, 2, 6, 4, 4, 8

Offset: 1

Gus Wiseman, Mar 08 2025

nonn

First differs from A355733 and A355735 at a(21) = 6, A355733(21) = A355735(21) = 5.
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 a(21) = 6 multisets are: {2,4}, {1,1,4}, {2,2,2}, {1,1,2,2}, {2,1,1,1,1}, {1,1,1,1,1,1}.
The a(n) partitions for n = 1, 3, 7, 13, 53, 21 (G = 16):
  ()  (2)   (4)     (6)       (G)                 (42)
      (11)  (22)    (33)      (88)                (411)
            (1111)  (222)     (4444)              (222)
                    (111111)  (22222222)          (2211)
                              (1111111111111111)  (21111)
                                                  (111111)

Robert Price, Table of n, a(n) for n = 1..300

Positions of 1 are A000079.
The strict case is A008966.
Before sorting we had A355731.
Choosing divisors instead of constant multisets gives A355733.
The upper version is A381455, before taking sums A000688.
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, zeros A293243) see A381441 (upper).
- For sets of constant multisets (A050361) see A381715.
- For strict multiset partitions with distinct sums (A321469) see A381637.
- For set systems with distinct sums (A381633, zeros A381806) see A381634.
- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.
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, A000961, A001222, A002577, A018818, A213242, A213385, A213427, A275870, A299200, A300273, A300385.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Sort/@Join@@@Tuples[Select[IntegerPartitions[#],SameQ@@#&]&/@prix[n]]]],{n,nn}]

a(A002110(n)) = A381807(n).

cp's OEIS Frontend

A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

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A055887 Number of ordered partitions of partitions.

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A050361 Number of factorizations into distinct prime powers greater than 1.

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A023893 Number of partitions of n into prime power parts (1 included); number of nonisomorphic Abelian subgroups of symmetric group S_n.

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A023894 Number of partitions of n into prime power parts (1 excluded).

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A381636 Numbers whose prime indices cannot be partitioned into constant blocks with distinct sums.

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A381635 Number of ways to partition the prime indices of n into constant blocks with distinct sums.

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