A372887 Number of integer partitions of n whose distinct parts are the binary indices of some prime number.

0, 0, 1, 1, 3, 3, 6, 8, 12, 14, 21, 29, 36, 48, 56, 74, 94, 123, 144, 195, 235, 301, 356, 456, 538, 679, 803, 997, 1189, 1467, 1716, 2103, 2488, 2968, 3517, 4185, 4907, 5834, 6850, 8032, 9459, 11073, 12933, 15130, 17652, 20480, 24011, 27851, 32344, 37520

Offset

0,5

Comments

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

Note the inverse of A048793 (binary indices) takes a set s to Sum_i 2^(s_i-1).

Links

Table of n, a(n) for n=0..49.

Example

The partition y = (4,3,1,1) has distinct parts {1,3,4}, which are the binary indices of 13, which is prime, so y is counted under a(9).
The a(2) = 1 through a(9) = 14 partitions:
  (2)  (21)  (22)   (221)   (51)     (331)     (431)      (3321)
             (31)   (311)   (222)    (421)     (521)      (4221)
             (211)  (2111)  (321)    (511)     (2222)     (4311)
                            (2211)   (2221)    (3221)     (5211)
                            (3111)   (3211)    (3311)     (22221)
                            (21111)  (22111)   (4211)     (32211)
                                     (31111)   (5111)     (33111)
                                     (211111)  (22211)    (42111)
                                               (32111)    (51111)
                                               (221111)   (222111)
                                               (311111)   (321111)
                                               (2111111)  (2211111)
                                                          (3111111)
                                                          (21111111)

Mathematica

Table[Length[Select[IntegerPartitions[n], PrimeQ[Total[2^(Union[#]-1)]]&]], {n, 0, 30}]

Crossrefs

For odd instead of prime we have A000041, even A002865.

The strict case is A372687, ranks A372851.

Counting not just distinct parts gives A372688, ranks A277319.

These partitions have Heinz numbers A372850.

A014499 lists binary indices of prime numbers.

A019565 gives Heinz number of binary indices, adjoint A048675.

A058698 counts partitions of prime numbers, strict A064688.

A372689 lists numbers whose binary indices sum to a prime.

A372885 lists primes whose binary indices sum to a prime, indices A372886.

Binary indices:

- listed A048793, sum A029931

- reversed A272020

- opposite A371572, sum A230877

- length A000120, complement A023416

- min A001511, opposite A000012

- max A070939, opposite A070940

- complement A368494, sum A359400

- opposite complement A371571, sum A359359

Cf. A000040, A000041, A029837, A035100, A038499, A372429, A372471.

Sequence in context: A241343 A309455 A168637 * A241390 A241831 A239946

Adjacent sequences: A372884 A372885 A372886 * A372888 A372889 A372890

Keyword

nonn

Author

Gus Wiseman, May 19 2024

Status

approved