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

%I A326958 #22 Nov 17 2020 04:43:44
%S A326958 0,1,4,9,16,31,52,87,132,203,303,450,641,922,1287,1792,2446,3347,4488,
%T A326958 6030,7975,10538,13778,17987,23234,29980,38383,49015,62195,78766,
%U A326958 99137,124560,155672,194158,241104,298780,368747,454276,557619,683132,834252,1016955
%N A326958 Total sum of noncomposite parts in all partitions of n.
%F A326958 a(n) = A073118(n) + A000070(n-1), n >= 1.
%F A326958 a(n) = A066186(n) - A326982(n).
%e A326958 For n = 6 we have:
%e A326958 --------------------------------------
%e A326958 .                         Sum of
%e A326958 Partitions             noncomposite
%e A326958 of 6                       parts
%e A326958 --------------------------------------
%e A326958 6 .......................... 0
%e A326958 3 + 3 ...................... 6
%e A326958 4 + 2 ...................... 2
%e A326958 2 + 2 + 2 .................. 6
%e A326958 5 + 1 ...................... 6
%e A326958 3 + 2 + 1 .................. 6
%e A326958 4 + 1 + 1 .................. 2
%e A326958 2 + 2 + 1 + 1 .............. 6
%e A326958 3 + 1 + 1 + 1 .............. 6
%e A326958 2 + 1 + 1 + 1 + 1 .......... 6
%e A326958 1 + 1 + 1 + 1 + 1 + 1 ...... 6
%e A326958 ------------------------------------
%e A326958 Total ..................... 52
%e A326958 So a(6) = 52.
%p A326958 b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], b(n, i-1)+
%p A326958       (p-> p+[0, `if`(isprime(i), p[1]*i, 0)])(b(n-i, min(n-i, i))))
%p A326958     end:
%p A326958 a:= n-> b(n$2)[2]:
%p A326958 seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 13 2019
%t A326958 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]]]];
%t A326958 a[n_] := b[n, n][[2]];
%t A326958 a /@ Range[0, 50] (* _Jean-François Alcover_, Nov 17 2020, after _Alois P. Heinz_ *)
%Y A326958 Cf. A000041, A000070, A008578 (noncomposites), A066186, A073118, A194545, A199936, A326957, A326982.
%K A326958 nonn
%O A326958 0,3
%A A326958 _Omar E. Pol_, Aug 08 2019