A344789 Number of partitions of the n-th nonprime number into a nonprime number of nonprime parts.

1, 2, 2, 2, 4, 3, 6, 8, 11, 11, 19, 27, 32, 37, 55, 63, 78, 88, 108, 149, 204, 232, 274, 313, 371, 497, 556, 654, 864, 1135, 1267, 1476, 1915, 2142, 2474, 2754, 3182, 4070, 4528, 5190, 5769, 6594, 8347, 10530, 11666, 13240, 14657, 16597, 20747, 22924, 25854

Offset: 1

Author

Alois P. Heinz, May 28 2021

Keywords

• nonn

Examples

			a(5) = 4: [9], [6,1,1,1], [4,1,1,1,1,1], [1,1,1,1,1,1,1,1,1].
a(6) = 3: [10], [4,4,1,1], [1,1,1,1,1,1,1,1,1,1].
a(7) = 6: [12], [9,1,1,1], [6,4,1,1], [4,4,1,1,1,1], [4,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A018252, A316154.

Programs

Maple

c:= proc(n) option remember; local k; if n=1 then 1 else
      for k from 1+c(n-1) while isprime(k) do od; k fi
    end:
h:= proc(n) option remember; `if`(isprime(n), h(n-1), n) end:
b:= proc(n, i, c) option remember; `if`(n=0 or i=1, `if`(isprime(
      c+n), 0, 1), b(n-i, h(min(n-i, i)), c+1)+b(n, h(i-1), c))
    end:
a:= n-> b(c(n)$2, 0):
seq(a(n), n=1..55);

Mathematica

c[n_] := c[n] = Module[{k}, If[n == 1, 1,
   For[k = 1+c[n-1], PrimeQ[k], k++]; k]];
h[n_] := h[n] = If[PrimeQ[n], h[n-1], n];
b[n_, i_, c_] := b[n, i, c] = If[n == 0 || i == 1, If[PrimeQ[
   c+n], 0, 1], b[n-i, h[Min[n-i, i]], c+1] + b[n, h[i-1], c]];
a[n_] := b[c[n], c[n], 0];
Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Sep 08 2022, after Alois P. Heinz *)