A326982 Total sum of composite parts in all partitions of n.

0, 0, 0, 0, 4, 4, 14, 18, 44, 67, 117, 166, 283, 391, 603, 848, 1250, 1702, 2442, 3280, 4565, 6094, 8266, 10878, 14566, 18970, 24953, 32255, 41909, 53619, 68983, 87542, 111496, 140561, 177436, 222125, 278425, 346293, 430951, 533083, 659268, 810948, 997322

Offset: 0

Author

Omar E. Pol, Aug 09 2019

Keywords

• nonn

Examples

			For n = 6 we have:
--------------------------------------
Partitions                Sum of
of 6                  composite parts
--------------------------------------
6 .......................... 6
3 + 3 ...................... 0
4 + 2 ...................... 4
2 + 2 + 2 .................. 0
5 + 1 ...................... 0
3 + 2 + 1 .................. 0
4 + 1 + 1 .................. 4
2 + 2 + 1 + 1 .............. 0
3 + 1 + 1 + 1 .............. 0
2 + 1 + 1 + 1 + 1 .......... 0
1 + 1 + 1 + 1 + 1 + 1 ...... 0
--------------------------------------
Total ..................... 14
So a(6) = 14.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..8000

Crossrefs

Cf. A000041, A002808, A066186, A073118, A194544, A194545, A199936, A326958, A326981.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, 0], b(n, i-1)+
      (p-> p+[0, `if`(isprime(i), 0, p[1]*i)])(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

Table[Total[Select[Flatten[IntegerPartitions[n]],CompositeQ]],{n,0,50}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 19 2020 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, 0}, b[n, i - 1] +
     With[{p = b[n-i, Min[n-i, i]]}, p+{0, If[PrimeQ[i], 0, p[[1]]*i]}]];
a[n_] := b[n, n][[2]];
a /@ Range[0, 50] (* Jean-François Alcover, Jun 07 2021, after Alois P. Heinz *)

Formula

a(n) = A194545(n) - A000070(n-1), n >= 1.

a(n) = A066186(n) - A326958(n).