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%I #18 Oct 29 2015 08:04:42
%S 1,2,4,8,13,22,34,54,80,119,170,246,342,478,653,894,1198,1610,2127,
%T 2813,3672,4789,6181,7975,10192,13010,16488,20861,26224,32918,41086,
%U 51199,63494,78599,96888,119235,146167,178879,218181,265662,322487,390834,472343
%N Total number of nonprime parts in all partitions of n.
%C a(n) is also the sum of the differences between the sum of m-th largest and the sum of (m+1)st largest elements in all partitions of n for all nonprimes m. - _Omar E. Pol_, Oct 27 2012
%H Alois P. Heinz, <a href="/A144119/b144119.txt">Table of n, a(n) for n = 1..1000</a>
%F a(n) = A006128(n)-A037032(n).
%e From _Omar E. Pol_, Nov 20 2011 (Start):
%e For n = 6 we have:
%e --------------------------------------
%e . Number of
%e Partitions nonprime parts
%e --------------------------------------
%e 6 .......................... 1
%e 3 + 3 ...................... 0
%e 4 + 2 ...................... 1
%e 2 + 2 + 2 .................. 0
%e 5 + 1 ...................... 1
%e 3 + 2 + 1 .................. 1
%e 4 + 1 + 1 .................. 3
%e 2 + 2 + 1 + 1 .............. 2
%e 3 + 1 + 1 + 1 .............. 3
%e 2 + 1 + 1 + 1 + 1 .......... 4
%e 1 + 1 + 1 + 1 + 1 + 1 ...... 6
%e ------------------------------------
%e Total ..................... 22
%e So a(6) = 22. (End)
%p b:= proc(n, i) option remember; local g;
%p if n=0 then [1, 0]
%p elif i<1 then [0, 0]
%p else g:= `if`(i>n, [0$2], b(n-i, i));
%p b(n, i-1) +g +[0, `if`(isprime(i), 0, g[1])]
%p fi
%p end:
%p a:= n-> b(n, n)[2]:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Oct 27 2012
%t b[n_, i_] := b[n, i] = Module[{g}, If[n == 0, {1, 0}, If[i<1, {0, 0}, g = If[i>n, {0, 0}, b[n-i, i]]; b[n, i-1] + g + {0, If[PrimeQ[i], 0, g[[1]]]} ]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Oct 29 2015, after _Alois P. Heinz_ *)
%o (PARI) vector(100, n, sum(k=1, n, (numdiv(k)-omega(k))*numbpart(n-k))) \\ _Altug Alkan_, Oct 29 2015
%Y Cf. A006128, A018252, A037032, A144116, A144121.
%K easy,nonn
%O 1,2
%A _Omar E. Pol_, Sep 11 2008
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