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 A073118 Total sum of prime parts in all partitions of n. 10
 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 (list; graph; refs; listen; history; text; internal format)
 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 A014495 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

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Last modified March 30 09:13 EDT 2020. Contains 333125 sequences. (Running on oeis4.)