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

%I A073118 #31 Aug 07 2019 15:50:39
%S A073118 0,2,5,9,19,33,57,87,136,206,311,446,650,914,1284,1762,2432,3276,4433,
%T A073118 5888,7824,10272,13479,17471,22642,29087,37283,47453,60306,76112,
%U A073118 95931,120201,150338,187141,232507,287591,355143,436849,536347,656282,801647,976095
%N A073118 Total sum of prime parts in all partitions of n.
%H A073118 Alois P. Heinz, <a href="/A073118/b073118.txt">Table of n, a(n) for n = 1..1000</a>
%F A073118 a(n) = Sum_{k=1..n} A008472(k)*A000041(n-k).
%F A073118 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
%e A073118 From _Omar E. Pol_, Nov 20 2011 (Start):
%e A073118 For n = 6 we have:
%e A073118 --------------------------------------
%e A073118 .                          Sum of
%e A073118 Partitions              prime parts
%e A073118 --------------------------------------
%e A073118 6 .......................... 0
%e A073118 3 + 3 ...................... 6
%e A073118 4 + 2 ...................... 2
%e A073118 2 + 2 + 2 .................. 6
%e A073118 5 + 1 ...................... 5
%e A073118 3 + 2 + 1 .................. 5
%e A073118 4 + 1 + 1 .................. 0
%e A073118 2 + 2 + 1 + 1 .............. 4
%e A073118 3 + 1 + 1 + 1 .............. 3
%e A073118 2 + 1 + 1 + 1 + 1 .......... 2
%e A073118 1 + 1 + 1 + 1 + 1 + 1 ...... 0
%e A073118 --------------------------------------
%e A073118 Total ..................... 33
%e A073118 So a(6) = 33. (End)
%p A073118 b:= proc(n, i) option remember; local h, j, t;
%p A073118       if n<0 then [0, 0]
%p A073118     elif n=0 then [1, 0]
%p A073118     elif i<1 then [0, 0]
%p A073118     else h:= [0, 0];
%p A073118          for j from 0 to iquo(n, i) do
%p A073118            t:= b(n-i*j, i-1);
%p A073118            h:= [h[1]+t[1], h[2]+t[2]+`if`(isprime(i), t[1]*i*j, 0)]
%p A073118          od; h
%p A073118       fi
%p A073118     end:
%p A073118 a:= n-> b(n, n)[2]:
%p A073118 seq(a(n), n=1..50);  # _Alois P. Heinz_, Nov 20 2011
%t A073118 f[n_] := Apply[Plus, Select[ Flatten[ IntegerPartitions[n]], PrimeQ[ # ] & ]]; Table[ f[n], {n, 1, 41} ]
%t A073118 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 *)
%o A073118 (PARI) a(n)={sum(k=1, n, vecsum(factor(k)[, 1])*numbpart(n-k))} \\ _Andrew Howroyd_, Dec 28 2017
%Y A073118 Cf. A037032, A194545, A309561.
%K A073118 easy,nonn
%O A073118 1,2
%A A073118 _Vladeta Jovovic_, Aug 24 2002
%E A073118 Edited and extended by _Robert G. Wilson v_, Aug 26 2002