OFFSET
0,2
COMMENTS
Primes of parts means the prime counting function applied to the part sizes. Equivalently, a(n) is the number of integer partitions with part sizes in A014689(n) interpreted as a multiset. - Andrew Howroyd, Apr 17 2021
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1/Product_{k>=1} 1 - x^(prime(k)-k). - Andrew Howroyd, Apr 16 2021
EXAMPLE
The a(0) = 1 through a(5) = 16 partitions:
() (1) (3) (4) (33) (43)
(2) (11) (31) (41) (331)
(21) (32) (42) (332)
(22) (111) (311) (411)
(211) (321) (421)
(221) (322) (422)
(222) (1111) (3111)
(2111) (3211)
(2211) (3221)
(2221) (3222)
(2222) (11111)
(21111)
(22111)
(22211)
(22221)
(22222)
For example, the partition (3,2,2,1) is counted under n = 5 because it has sum of primes 5+3+3+2 = 13 and its sum of parts plus n is also 3+2+2+1+5 = 13.
MATHEMATICA
Table[Sum[Length[Select[IntegerPartitions[k], Total[Prime/@#]==k+n&]], {k, 0, 2*n}], {n, 0, 10}]
PROG
(PARI) seq(n)={my(m=1); while(prime(m)-m<=n, m++); Vec(1/prod(k=1, m, 1 - x^(prime(k)-k) + O(x*x^n)))} \\ Andrew Howroyd, Apr 16 2021
CROSSREFS
Column sums of A331385.
Partitions into primes are A000607.
Partitions whose sum of primes is divisible by their sum are A331379.
Partitions whose product divides their sum of primes are A331381.
Partitions whose product equals their sum of primes are A331383.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 17 2020
EXTENSIONS
Terms a(31) and beyond from Andrew Howroyd, Apr 16 2021
STATUS
approved