A073118 Total sum of prime parts in all partitions of n.

0, 2, 5, 9, 19, 33, 57, 87, 136, 206, 311, 446, 650, 914, 1284, 1762, 2432, 3276, 4433, 5888, 7824, 10272, 13479, 17471, 22642, 29087, 37283, 47453, 60306, 76112, 95931, 120201, 150338, 187141, 232507, 287591, 355143, 436849, 536347, 656282, 801647, 976095

Offset

1,2

Links

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

Formula

a(n) = Sum_{k=1..n} A008472(k)*A000041(n-k).

G.f.: Sum_{i>=1} prime(i)*x^prime(i)/(1 - x^prime(i)) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Feb 01 2017

Example

From Omar E. Pol, Nov 20 2011 (Start):
For n = 6 we have:
--------------------------------------
.                          Sum of
Partitions              prime parts
--------------------------------------
6 .......................... 0
3 + 3 ...................... 6
4 + 2 ...................... 2
2 + 2 + 2 .................. 6
5 + 1 ...................... 5
3 + 2 + 1 .................. 5
4 + 1 + 1 .................. 0
2 + 2 + 1 + 1 .............. 4
3 + 1 + 1 + 1 .............. 3
2 + 1 + 1 + 1 + 1 .......... 2
1 + 1 + 1 + 1 + 1 + 1 ...... 0
--------------------------------------
Total ..................... 33
So a(6) = 33. (End)

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), t[1]*i*j, 0)]
         od; h
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Nov 20 2011

Mathematica

f[n_] := Apply[Plus, Select[ Flatten[ IntegerPartitions[n]], PrimeQ[ # ] & ]]; Table[ f[n], {n, 1, 41} ]
a[n_] := Sum[Total[FactorInteger[k][[All, 1]]]*PartitionsP[n-k], {k, 1, n}] - PartitionsP[n-1]; Array[a, 50] (* Jean-François Alcover, Dec 27 2015 *)

Prog

(PARI) a(n)={sum(k=1, n, vecsum(factor(k)[, 1])*numbpart(n-k))} \\ Andrew Howroyd, Dec 28 2017

Crossrefs

Cf. A037032, A194545, A309561.

Sequence in context: A213544 A265482 A085410 * A048082 A089089 A369854

Adjacent sequences: A073115 A073116 A073117 * A073119 A073120 A073121

Keyword

easy,nonn

Author

Vladeta Jovovic, Aug 24 2002

Extensions

Edited and extended by Robert G. Wilson v, Aug 26 2002

Status

approved