A325622 Number of integer partitions of n whose reciprocal factorial sum is the reciprocal of an integer.

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 3, 3, 2, 2, 3, 3, 3, 5, 4, 4, 3, 3, 4, 6, 3, 4, 5, 5, 5, 6, 3, 7, 6, 5, 6, 6, 6, 5, 6, 8, 5, 7, 5, 4, 8, 7, 7, 7, 7, 9, 9, 9, 10, 12, 6, 12, 8, 10, 7, 14, 10, 8, 11, 11, 12, 11, 10, 10, 12, 14

Offset

1,4

Comments

The reciprocal factorial sum of an integer partition (y_1,...,y_k) is 1/y_1! + ... + 1/y_k!.

Links

Table of n, a(n) for n=1..70.

Example

The initial terms count the following partitions:
   1: (1)
   2: (2)
   3: (3)
   4: (4)
   4: (2,2)
   5: (5)
   6: (6)
   6: (3,3)
   7: (7)
   8: (8)
   8: (4,4)
   9: (9)
   9: (5,4)
   9: (3,3,3)
  10: (10)
  10: (5,5)
  11: (11)
  11: (4,4,3)
  11: (3,3,3,2)
  12: (12)
  12: (6,6)
  12: (4,4,4)

Maple

f:= proc(n) nops(select(proc(t) local i; (1/add(1/i!, i=t))::integer end proc, combinat:-partition(n))) end proc:
map(f, [$1..70]); # Robert Israel, May 09 2024

Mathematica

Table[Length[Select[IntegerPartitions[n], IntegerQ[1/Total[1/(#!)]]&]], {n, 30}]

Crossrefs

Factorial numbers: A000142, A007489, A022559, A064986, A108731, A115944, A284605, A325508, A325616.

Reciprocal factorial sum: A002966, A316854, A316857, A325618, A325620, A325623.

Sequence in context: A194509 A054716 A261641 * A060145 A358997 A373461

Adjacent sequences: A325619 A325620 A325621 * A325623 A325624 A325625

Keyword

nonn,more

Author

Gus Wiseman, May 13 2019

Extensions

a(61)-a(70) from Robert Israel, May 09 2024

Status

approved