A326958 Total sum of noncomposite parts in all partitions of n.

0, 1, 4, 9, 16, 31, 52, 87, 132, 203, 303, 450, 641, 922, 1287, 1792, 2446, 3347, 4488, 6030, 7975, 10538, 13778, 17987, 23234, 29980, 38383, 49015, 62195, 78766, 99137, 124560, 155672, 194158, 241104, 298780, 368747, 454276, 557619, 683132, 834252, 1016955

Offset

0,3

Links

Table of n, a(n) for n=0..41.

Formula

a(n) = A073118(n) + A000070(n-1), n >= 1.

a(n) = A066186(n) - A326982(n).

Example

For n = 6 we have:
--------------------------------------
.                         Sum of
Partitions             noncomposite
of 6                       parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 6
4 + 2 ...................... 2
2 + 2 + 2 .................. 6
5 + 1 ...................... 6
3 + 2 + 1 .................. 6
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 6
3 + 1 + 1 + 1 .............. 6
2 + 1 + 1 + 1 + 1 .......... 6
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
Total ..................... 52
So a(6) = 52.

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], b(n, i-1)+
      (p-> p+[0, `if`(isprime(i), p[1]*i, 0)])(b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 13 2019

Mathematica

b[n_, i_] := b[n, i] = If[n==0 || i==1, {1, n}, b[n, i-1] + # + {0, If[PrimeQ[i], #[[1]] i, 0]}&[b[n-i, Min[n-i, i]]]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 50] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

Crossrefs

Cf. A000041, A000070, A008578 (noncomposites), A066186, A073118, A194545, A199936, A326957, A326982.

Sequence in context: A093175 A138992 A199936 * A281904 A007679 A239870

Adjacent sequences: A326955 A326956 A326957 * A326959 A326960 A326961

Keyword

nonn

Author

Omar E. Pol, Aug 08 2019

Status

approved