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

%I A326981 #17 Nov 17 2020 04:45:01
%S A326981 0,0,0,0,1,1,3,4,9,13,22,31,51,70,105,145,210,283,398,530,726,958,
%T A326981 1283,1673,2212,2854,3714,4756,6119,7764,9893,12457,15728,19674,24636,
%U A326981 30615,38079,47034,58109,71396,87692,107179,130943,159278,193619,234486,283720
%N A326981 Total number of composite parts in all partitions of n.
%F A326981 a(n) = A144119(n) - A000070(n-1), n >= 1.
%F A326981 a(n) = A006128(n) - A326957(n).
%e A326981 For n = 6 we have:
%e A326981 --------------------------------------
%e A326981 .                        Number of
%e A326981 Partitions               composite
%e A326981 of 6                       parts
%e A326981 --------------------------------------
%e A326981 6 .......................... 1
%e A326981 3 + 3 ...................... 0
%e A326981 4 + 2 ...................... 1
%e A326981 2 + 2 + 2 .................. 0
%e A326981 5 + 1 ...................... 0
%e A326981 3 + 2 + 1 .................. 0
%e A326981 4 + 1 + 1 .................. 1
%e A326981 2 + 2 + 1 + 1 .............. 0
%e A326981 3 + 1 + 1 + 1 .............. 0
%e A326981 2 + 1 + 1 + 1 + 1 .......... 0
%e A326981 1 + 1 + 1 + 1 + 1 + 1 ...... 0
%e A326981 ------------------------------------
%e A326981 Total ...................... 3
%e A326981 So a(6) = 3.
%p A326981 b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, 0], b(n, i-1)+
%p A326981       (p-> p+[0, `if`(isprime(i), 0, p[1])])(b(n-i, min(n-i, i))))
%p A326981     end:
%p A326981 a:= n-> b(n$2)[2]:
%p A326981 seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 13 2019
%t A326981 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]]]];
%t A326981 a[n_] := b[n, n][[2]];
%t A326981 a /@ Range[0, 50] (* _Jean-François Alcover_, Nov 17 2020, after _Alois P. Heinz_ *)
%Y A326981 Cf. A000041, A002808, A006128, A037032, A144115, A144116, A144119, A326957, A326982.
%K A326981 nonn
%O A326981 0,7
%A A326981 _Omar E. Pol_, Aug 09 2019