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
Examples
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)
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000
Programs
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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 *) -
PARI
a(n)={sum(k=1, n, vecsum(factor(k)[, 1])*numbpart(n-k))} \\ Andrew Howroyd, Dec 28 2017
Formula
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
Extensions
Edited and extended by Robert G. Wilson v, Aug 26 2002