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A326957 Total number of noncomposite parts in all partitions of n.

%I A326957 #22 Nov 15 2020 08:58:32
%S A326957 0,1,3,6,11,19,32,50,77,115,170,244,348,486,675,923,1253,1682,2246,
%T A326957 2968,3904,5094,6616,8533,10962,13997,17808,22538,28426,35689,44670,
%U A326957 55678,69199,85692,105826,130261,159935,195778,239092,291191,353854,428925,518848
%N A326957 Total number of noncomposite parts in all partitions of n.
%F A326957 a(n) = A037032(n) + A000070(n-1), n >= 1.
%F A326957 a(n) = A006128(n) - A326981(n).
%e A326957 For n = 6 we have:
%e A326957 --------------------------------------
%e A326957 .                        Number of
%e A326957 Partitions             noncomposite
%e A326957 of 6                       parts
%e A326957 --------------------------------------
%e A326957 6 .......................... 0
%e A326957 3 + 3 ...................... 2
%e A326957 4 + 2 ...................... 1
%e A326957 2 + 2 + 2 .................. 3
%e A326957 5 + 1 ...................... 2
%e A326957 3 + 2 + 1 .................. 3
%e A326957 4 + 1 + 1 .................. 2
%e A326957 2 + 2 + 1 + 1 .............. 4
%e A326957 3 + 1 + 1 + 1 .............. 4
%e A326957 2 + 1 + 1 + 1 + 1 .......... 5
%e A326957 1 + 1 + 1 + 1 + 1 + 1 ...... 6
%e A326957 ------------------------------------
%e A326957 Total ..................... 32
%e A326957 So a(6) = 32.
%p A326957 b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], b(n, i-1)+
%p A326957       (p-> p+[0, `if`(isprime(i), p[1], 0)])(b(n-i, min(n-i, i))))
%p A326957     end:
%p A326957 a:= n-> b(n$2)[2]:
%p A326957 seq(a(n), n=0..50);  # _Alois P. Heinz_, Aug 13 2019
%t A326957 b[n_] := Sum[PrimeNu[k] PartitionsP[n-k], {k, 1, n}];
%t A326957 c[n_] := SeriesCoefficient[Product[1/(1-x^k), {k, 1, n}]/(1-x), {x, 0, n}];
%t A326957 a[n_] := b[n] + c[n-1];
%t A326957 a /@ Range[0, 50] (* _Jean-François Alcover_, Nov 15 2020 *)
%Y A326957 First differs from A183088 at a(13).
%Y A326957 Cf. A000041, A000070, A006128, A008578 (noncomposites), A037032, A144115, A144116, A144119, A326958, A326981.
%K A326957 nonn
%O A326957 0,3
%A A326957 _Omar E. Pol_, Aug 08 2019