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

0, 1, 2, 4, 11, 16, 33, 48, 89, 134, 214, 305, 478, 663, 976, 1356, 1934, 2617, 3654, 4877, 6652, 8808, 11772, 15386, 20329, 26308, 34249, 43987, 56651, 72079, 92008, 116171, 146967, 184381, 231399, 288398, 359581, 445426, 551721, 679868, 837238, 1026256

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = A066186(n) - A073118(n).

Example

For n = 6 we have:
--------------------------------------
.                          Sum of
Partitions             nonprime parts
--------------------------------------
6 .......................... 6
3 + 3 ...................... 0
4 + 2 ...................... 4
2 + 2 + 2 .................. 0
5 + 1 ...................... 1
3 + 2 + 1 .................. 1
4 + 1 + 1 .................. 6
2 + 2 + 1 + 1 .............. 2
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
--------------------------------------
Total ..................... 33
So a(6) = 33.

Maple

b:= proc(n, i) option remember; local h, j, t;
      if n<0 then [0, 0]
    elif n=0 then [1, 0]
    elif i<1 then [0, 0]
    else h:= [0, 0];
         for j from 0 to iquo(n, i) do
           t:= b(n-i*j, i-1);
           h:= [h[1]+t[1], h[2]+t[2]+`if`(isprime(i), 0, t[1]*i*j)]
         od; h
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 20 2011

Mathematica

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 *)

Crossrefs

Cf. A018252, A066186, A073118, A194544.

Sequence in context: A134419 A342228 A024819 * A329800 A278346 A277867

Adjacent sequences: A194542 A194543 A194544 * A194546 A194547 A194548

Keyword

nonn

Author

Omar E. Pol, Nov 20 2011

Extensions

More terms from Alois P. Heinz, Nov 20 2011

Status

approved