A305937 Number of partitions such that the least positive integer which is not a part of the partition is prime.

0, 1, 1, 2, 3, 5, 6, 10, 13, 19, 26, 36, 47, 65, 84, 111, 144, 188, 239, 309, 390, 497, 624, 786, 978, 1224, 1513, 1875, 2306, 2839, 3469, 4246, 5162, 6279, 7600, 9196, 11077, 13344, 16006, 19191, 22934, 27387, 32602, 38788, 46015, 54547, 64504, 76209, 89835

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Example

For n=3 there are three unrestricted partitions: 3, 2+1, and 1+1+1. The least positive integer not in the first partition is 1. One is not a prime so the first partition is not counted. For the second partition the smallest missing positive integer is 3, which is prime. For the third partition the missing number is 2, which is prime. So a(3)=2.

Maple

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t or isprime(i), 1, 0), `if`(i>n, 0,
      `if`(t or isprime(i), b(n, i+1, true), 0)+
       add(b(n-i*j, i+1, t), j=1..n/i)))
    end:
a:= n-> b(n, 1, false):
seq(a(n), n=0..70);  # Alois P. Heinz, Jun 16 2018

Mathematica

nend = 30;
For[n = 1, n <= nend, n++,
  sum[n] = 0;
  partition = {n};
  For[i = 1, i <= PartitionsP[n], i++,
   partition = NextPartition[partition];
   mex = Min[Complement[Range[n + 1], partition]];
   If [PrimeQ[mex], sum[n]++; ]    ] ];
Table[sum[i], {i, 1, nend}]

Crossrefs

Cf. A000040.

Sequence in context: A064173 A145724 A039843 * A286097 A287483 A014853

Adjacent sequences: A305934 A305935 A305936 * A305938 A305939 A305940

Keyword

nonn

Author

David S. Newman, Jun 14 2018

Status

approved