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

1, 2, 4, 8, 13, 22, 34, 54, 80, 119, 170, 246, 342, 478, 653, 894, 1198, 1610, 2127, 2813, 3672, 4789, 6181, 7975, 10192, 13010, 16488, 20861, 26224, 32918, 41086, 51199, 63494, 78599, 96888, 119235, 146167, 178879, 218181, 265662, 322487, 390834, 472343

Offset: 1

Author

Omar E. Pol, Sep 11 2008

Keywords

• easy
• nonn

Comments

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

Examples

			From _Omar E. Pol_, Nov 20 2011 (Start):
For n = 6 we have:
--------------------------------------
.                        Number of
Partitions            nonprime parts
--------------------------------------
6 .......................... 1
3 + 3 ...................... 0
4 + 2 ...................... 1
2 + 2 + 2 .................. 0
5 + 1 ...................... 1
3 + 2 + 1 .................. 1
4 + 1 + 1 .................. 3
2 + 2 + 1 + 1 .............. 2
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 4
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
Total ..................... 22
So a(6) = 22. (End)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A006128, A018252, A037032, A144116, A144121.

Programs

Maple

b:= proc(n, i) option remember; local g;
      if n=0 then [1, 0]
    elif i<1 then [0, 0]
    else g:= `if`(i>n, [0$2], b(n-i, i));
         b(n, i-1) +g +[0, `if`(isprime(i), 0, g[1])]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012

Mathematica

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 *)

PARI

vector(100, n, sum(k=1, n, (numdiv(k)-omega(k))*numbpart(n-k))) \\ Altug Alkan, Oct 29 2015

Formula

a(n) = A006128(n)-A037032(n).