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

Author

G. L. Honaker, Jr.

Keywords

• nonn

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

Examples

			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)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000041, A001221, A073118.

Programs

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 *)

PARI

a(n)={sum(k=1, n, omega(k)*numbpart(n-k))} \\ Andrew Howroyd, Dec 28 2017

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

Extensions

More terms from Naohiro Nomoto, Apr 19 2002