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 A037032 Total number of prime parts in all partitions of n. 14
 0, 1, 2, 4, 7, 13, 20, 32, 48, 73, 105, 153, 214, 302, 415, 569, 767, 1034, 1371, 1817, 2380, 3110, 4025, 5199, 6659, 8512, 10806, 13684, 17229, 21645, 27049, 33728, 41872, 51863, 63988, 78779, 96645, 118322, 144406, 175884, 213617, 258957, 313094, 377867 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS a(n) is also the sum of the differences between the sum of p-th largest and the sum of (p+1)st largest elements in all partitions of n for all primes p. - Omar E. Pol, Oct 25 2012 LINKS Alois P. Heinz, Table of n, a(n) for n = 1..1000 FORMULA a(n) = Sum_{k=1..n} A001221(k)*A000041(n-k). - Vladeta Jovovic, Aug 22 2002 a(n) = Sum_{k=1..floor(n/2)} k*A222656(n,k). - Alois P. Heinz, May 29 2013 G.f.: Sum_{i>=1} x^prime(i)/(1 - x^prime(i)) / Product_{j>=1} (1 - x^j). - Ilya Gutkovskiy, Jan 24 2017 EXAMPLE From Omar E. Pol, Nov 20 2011 (Start): For n = 6 we have: -------------------------------------- .                        Number of Partitions              prime parts -------------------------------------- 6 .......................... 0 3 + 3 ...................... 2 4 + 2 ...................... 1 2 + 2 + 2 .................. 3 5 + 1 ...................... 1 3 + 2 + 1 .................. 2 4 + 1 + 1 .................. 0 2 + 2 + 1 + 1 .............. 2 3 + 1 + 1 + 1 .............. 1 2 + 1 + 1 + 1 + 1 .......... 1 1 + 1 + 1 + 1 + 1 + 1 ...... 0 ------------------------------------ Total ..................... 13 So a(6) = 13. (End) MAPLE with(combinat): a:=proc(n) local P, c, j, i: P:=partition(n): c:=0: for j from 1 to numbpart(n) do for i from 1 to nops(P[j]) do if isprime(P[j][i])=true then c:=c+1 else c:=c fi: od: od: c: end: seq(a(n), n=1..42); # Emeric Deutsch, Mar 30 2006 # second Maple program b:= proc(n, i) option remember; local g;       if n=0 or i=1 then [1, 0]     else g:= `if`(i>n, [0\$2], b(n-i, i));          b(n, i-1) +g +[0, `if`(isprime(i), g[1], 0)]       fi     end: a:= n-> b(n, n)[2]: seq(a(n), n=1..100);  # Alois P. Heinz, Oct 27 2012 MATHEMATICA a[n_] := Sum[PrimeNu[k]*PartitionsP[n - k], {k, 1, n}]; Array[a, 100] (* Jean-François Alcover, Mar 16 2015, after Vladeta Jovovic *) PROG (PARI) a(n)={sum(k=1, n, omega(k)*numbpart(n-k))} \\ Andrew Howroyd, Dec 28 2017 CROSSREFS Cf. A000041, A001221, A073118. Sequence in context: A164901 A262744 A112997 * A165753 A266650 A205183 Adjacent sequences:  A037029 A037030 A037031 * A037033 A037034 A037035 KEYWORD nonn AUTHOR EXTENSIONS More terms from Naohiro Nomoto, Apr 19 2002 STATUS approved

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Last modified July 25 23:39 EDT 2021. Contains 346294 sequences. (Running on oeis4.)