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%I #22 Jun 07 2021 04:46:30
%S 0,0,0,0,4,4,14,18,44,67,117,166,283,391,603,848,1250,1702,2442,3280,
%T 4565,6094,8266,10878,14566,18970,24953,32255,41909,53619,68983,87542,
%U 111496,140561,177436,222125,278425,346293,430951,533083,659268,810948,997322
%N Total sum of composite parts in all partitions of n.
%H Alois P. Heinz, <a href="/A326982/b326982.txt">Table of n, a(n) for n = 0..8000</a>
%F a(n) = A194545(n) - A000070(n-1), n >= 1.
%F a(n) = A066186(n) - A326958(n).
%e For n = 6 we have:
%e --------------------------------------
%e Partitions Sum of
%e of 6 composite parts
%e --------------------------------------
%e 6 .......................... 6
%e 3 + 3 ...................... 0
%e 4 + 2 ...................... 4
%e 2 + 2 + 2 .................. 0
%e 5 + 1 ...................... 0
%e 3 + 2 + 1 .................. 0
%e 4 + 1 + 1 .................. 4
%e 2 + 2 + 1 + 1 .............. 0
%e 3 + 1 + 1 + 1 .............. 0
%e 2 + 1 + 1 + 1 + 1 .......... 0
%e 1 + 1 + 1 + 1 + 1 + 1 ...... 0
%e --------------------------------------
%e Total ..................... 14
%e So a(6) = 14.
%p b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, 0], b(n, i-1)+
%p (p-> p+[0, `if`(isprime(i), 0, p[1]*i)])(b(n-i, min(n-i, i))))
%p end:
%p a:= n-> b(n$2)[2]:
%p seq(a(n), n=0..50); # _Alois P. Heinz_, Aug 13 2019
%t Table[Total[Select[Flatten[IntegerPartitions[n]],CompositeQ]],{n,0,50}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Apr 19 2020 *)
%t b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {1, 0}, b[n, i - 1] +
%t With[{p = b[n-i, Min[n-i, i]]}, p+{0, If[PrimeQ[i], 0, p[[1]]*i]}]];
%t a[n_] := b[n, n][[2]];
%t a /@ Range[0, 50] (* _Jean-François Alcover_, Jun 07 2021, after _Alois P. Heinz_ *)
%Y Cf. A000041, A002808, A066186, A073118, A194544, A194545, A199936, A326958, A326981.
%K nonn
%O 0,5
%A _Omar E. Pol_, Aug 09 2019
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