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

1,2

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

Links

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

Formula

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

Example

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)

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 *)

Prog

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

Crossrefs

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

Sequence in context: A134035 A351620 A078157 * A207033 A291553 A330153

Adjacent sequences: A144116 A144117 A144118 * A144120 A144121 A144122

Keyword

easy,nonn

Author

Omar E. Pol, Sep 11 2008

Status

approved