cp's OEIS Frontend

A194545 Total sum of nonprime parts in all partitions of n.

%I A194545 #15 Nov 03 2015 03:18:46
%S A194545 0,1,2,4,11,16,33,48,89,134,214,305,478,663,976,1356,1934,2617,3654,
%T A194545 4877,6652,8808,11772,15386,20329,26308,34249,43987,56651,72079,92008,
%U A194545 116171,146967,184381,231399,288398,359581,445426,551721,679868,837238,1026256
%N A194545 Total sum of nonprime parts in all partitions of n.
%H A194545 Alois P. Heinz, <a href="/A194545/b194545.txt">Table of n, a(n) for n = 0..1000</a>
%F A194545 a(n) = A066186(n) - A073118(n).
%e A194545 For n = 6 we have:
%e A194545 --------------------------------------
%e A194545 .                          Sum of
%e A194545 Partitions             nonprime parts
%e A194545 --------------------------------------
%e A194545 6 .......................... 6
%e A194545 3 + 3 ...................... 0
%e A194545 4 + 2 ...................... 4
%e A194545 2 + 2 + 2 .................. 0
%e A194545 5 + 1 ...................... 1
%e A194545 3 + 2 + 1 .................. 1
%e A194545 4 + 1 + 1 .................. 6
%e A194545 2 + 2 + 1 + 1 .............. 2
%e A194545 3 + 1 + 1 + 1 .............. 3
%e A194545 2 + 1 + 1 + 1 + 1 .......... 4
%e A194545 1 + 1 + 1 + 1 + 1 + 1 ...... 6
%e A194545 --------------------------------------
%e A194545 Total ..................... 33
%e A194545 So a(6) = 33.
%p A194545 b:= proc(n, i) option remember; local h, j, t;
%p A194545       if n<0 then [0, 0]
%p A194545     elif n=0 then [1, 0]
%p A194545     elif i<1 then [0, 0]
%p A194545     else h:= [0, 0];
%p A194545          for j from 0 to iquo(n, i) do
%p A194545            t:= b(n-i*j, i-1);
%p A194545            h:= [h[1]+t[1], h[2]+t[2]+`if`(isprime(i), 0, t[1]*i*j)]
%p A194545          od; h
%p A194545       fi
%p A194545     end:
%p A194545 a:= n-> b(n, n)[2]:
%p A194545 seq(a(n), n=0..50);  # _Alois P. Heinz_, Nov 20 2011
%t A194545 b[n_, i_] := b[n, i] = Module[{h, j, t}, Which[n<0, {0, 0}, n==0, {1, 0}, i < 1, {0, 0}, True, h = {0, 0}; For[j = 0, j <= Quotient[n, i], j++, t = b[n-i*j, i-1]; h = {h[[1]] + t[[1]], h[[2]] + t[[2]] + If[PrimeQ[i], 0, t[[1]]*i*j]}]; h]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 50}] (* _Jean-François Alcover_, Nov 03 2015, after _Alois P. Heinz_ *)
%Y A194545 Cf. A018252, A066186, A073118, A194544.
%K A194545 nonn
%O A194545 0,3
%A A194545 _Omar E. Pol_, Nov 20 2011
%E A194545 More terms from _Alois P. Heinz_, Nov 20 2011