|
|
A023893
|
|
Number of partitions of n into prime power parts (1 included); number of nonisomorphic Abelian subgroups of symmetric group S_n.
|
|
47
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
LINKS
|
|
|
FORMULA
|
G.f.: (Product_{p prime} Product_{k>=1} 1/(1-x^(p^k))) / (1-x).
|
|
EXAMPLE
|
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)
|
|
MATHEMATICA
|
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 *)
|
|
PROG
|
(PARI) 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
|
|
CROSSREFS
|
The multiplicative version (factorizations) is A000688.
The version for just primes (not prime-powers) is A034891, strict A036497.
These partitions are ranked by A302492.
A001222 counts prime-power divisors.
A072233 counts partitions by sum and length.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|