A326957 Total number of noncomposite parts in all partitions of n.

0, 1, 3, 6, 11, 19, 32, 50, 77, 115, 170, 244, 348, 486, 675, 923, 1253, 1682, 2246, 2968, 3904, 5094, 6616, 8533, 10962, 13997, 17808, 22538, 28426, 35689, 44670, 55678, 69199, 85692, 105826, 130261, 159935, 195778, 239092, 291191, 353854, 428925, 518848

Offset: 0

Author

Omar E. Pol, Aug 08 2019

Keywords

• nonn

Examples

			For n = 6 we have:
--------------------------------------
.                        Number of
Partitions             noncomposite
of 6                       parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 2
4 + 2 ...................... 1
2 + 2 + 2 .................. 3
5 + 1 ...................... 2
3 + 2 + 1 .................. 3
4 + 1 + 1 .................. 2
2 + 2 + 1 + 1 .............. 4
3 + 1 + 1 + 1 .............. 4
2 + 1 + 1 + 1 + 1 .......... 5
1 + 1 + 1 + 1 + 1 + 1 ...... 6
------------------------------------
Total ..................... 32
So a(6) = 32.
		

Crossrefs

First differs from A183088 at a(13).

Cf. A000041, A000070, A006128, A008578 (noncomposites), A037032, A144115, A144116, A144119, A326958, A326981.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i=1, [1, n], b(n, i-1)+
      (p-> p+[0, `if`(isprime(i), p[1], 0)])(b(n-i, min(n-i, i))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 13 2019

Mathematica

b[n_] := Sum[PrimeNu[k] PartitionsP[n-k], {k, 1, n}];
c[n_] := SeriesCoefficient[Product[1/(1-x^k), {k, 1, n}]/(1-x), {x, 0, n}];
a[n_] := b[n] + c[n-1];
a /@ Range[0, 50] (* Jean-François Alcover, Nov 15 2020 *)

Formula

a(n) = A037032(n) + A000070(n-1), n >= 1.

a(n) = A006128(n) - A326981(n).