A037032 Total number of prime parts in all partitions of n.

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

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: A359516 A347703 A112997 * A165753 A347779 A369520

Adjacent sequences: A037029 A037030 A037031 * A037033 A037034 A037035

Keyword

nonn

Author

G. L. Honaker, Jr.

Extensions

More terms from Naohiro Nomoto, Apr 19 2002

Status

approved