A326981 Total number of composite parts in all partitions of n.

0, 0, 0, 0, 1, 1, 3, 4, 9, 13, 22, 31, 51, 70, 105, 145, 210, 283, 398, 530, 726, 958, 1283, 1673, 2212, 2854, 3714, 4756, 6119, 7764, 9893, 12457, 15728, 19674, 24636, 30615, 38079, 47034, 58109, 71396, 87692, 107179, 130943, 159278, 193619, 234486, 283720

Offset

0,7

Links

Table of n, a(n) for n=0..46.

Formula

a(n) = A144119(n) - A000070(n-1), n >= 1.

a(n) = A006128(n) - A326957(n).

Example

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

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])])(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, 0}, b[n, i-1] + # + {0, If[PrimeQ[i], 0, #[[1]]]}&[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, A002808, A006128, A037032, A144115, A144116, A144119, A326957, A326982.

Sequence in context: A225288 A377040 A167930 * A124285 A131326 A089300

Adjacent sequences: A326978 A326979 A326980 * A326982 A326983 A326984

Keyword

nonn

Author

Omar E. Pol, Aug 09 2019

Status

approved