A332248 Number of set partitions of [n] where all prime-indexed blocks are not singletons.

1, 1, 1, 2, 5, 15, 60, 286, 1423, 7185, 37758, 212596, 1293577, 8415869, 57715274, 414520958, 3125102795, 24880061105, 209909409566, 1871945790360, 17503956383037, 169851122851049, 1694189515772750, 17248694322541778, 178473482993477591, 1873036127628583885

Offset: 0

Author

Alois P. Heinz, Feb 12 2020

Keywords

• nonn

Examples

			a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 123, 1|23.
a(4) = 5: 1234, 12|34, 13|24, 14|23, 1|234.
a(5) = 15: 12345, 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345, 1|23|45, 1|24|35, 1|25|34.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..576
• Wikipedia, Partition of a set

Crossrefs

Cf. A000040, A000110, A000296, A332398.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, add(b(n-j, i+1)*
       binomial(n-1, j-1), j=`if`(isprime(i), 2, 1)..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..32);

Mathematica

b[n_, i_] := b[n, i] = If[n==0, 1, Sum[b[n-j, i+1] Binomial[n-1, j-1], {j, If[PrimeQ[i], 2, 1], n}]];
a[n_] := b[n, 1];
a /@ Range[0, 32] (* Jean-François Alcover, May 08 2020, after Maple *)