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

0,5

Links

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

Formula

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

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

Example

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.

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 *)

Crossrefs

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

Sequence in context: A263870 A263796 A263871 * A263797 A376531 A331609

Adjacent sequences: A326979 A326980 A326981 * A326983 A326984 A326985

Keyword

nonn

Author

Omar E. Pol, Aug 09 2019

Status

approved