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A144119 Total number of nonprime parts in all partitions of n.

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