A331387 Number of integer partitions whose sum of primes of parts equals their sum of parts plus n.

1, 2, 4, 7, 11, 16, 24, 34, 47, 64, 86, 113, 148, 191, 245, 310, 390, 486, 602, 740, 907, 1104, 1338, 1613, 1937, 2315, 2758, 3272, 3871, 4562, 5362, 6283, 7344, 8558, 9952, 11542, 13356, 15419, 17766, 20425, 23440, 26846, 30696, 35032, 39917, 45406

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.

Cf. A000040, A001414, A014689, A056239, A330950, A330953, A330954, A331378, A331415, A331416, A331418.

Sequence in context: A011912 A063676 A099385 * A357308 A357933 A237821

Adjacent sequences: A331384 A331385 A331386 * A331388 A331389 A331390

Keyword

nonn

Author

Gus Wiseman, Jan 17 2020

Extensions

Terms a(31) and beyond from Andrew Howroyd, Apr 16 2021

Status

approved