A336632 -id:A336632 - cp's OEIS frontend

A102291 Total number of prime parts in all compositions of n.

0, 0, 1, 3, 7, 18, 42, 98, 222, 497, 1100, 2413, 5250, 11350, 24398, 52193, 111180, 235949, 499074, 1052502, 2213710, 4644833, 9724492, 20318637, 42376578, 88231765, 183420748, 380755932, 789340736, 1634339217, 3379993922, 6982618822, 14410499598, 29711523105

Offset: 0

Vladeta Jovovic, Feb 19 2005

Alois P. Heinz, Table of n, a(n) for n = 0..3313

Cf. A037032, A224344, A336632.

a:= proc(n) option remember; `if`(n=0, 0, add(a(n-j)+
      `if`(isprime(j), ceil(2^(n-j-1)), 0), j=1..n))
    end:
seq(a(n), n=0..33);  # Alois P. Heinz, Aug 06 2019

a[n_] := a[n] = If[n==0, 0, Sum[a[n-j] + If[PrimeQ[j], Ceiling[2^(n-j-1)], 0], {j, 1, n}]];
a /@ Range[0, 33] (* Jean-François Alcover, Oct 30 2020, after Alois P. Heinz *)

G.f.: Sum_{k>=1} x^prime(k)*(1-x)^2/(1-2*x)^2.

a(n) = Sum_{k=1..floor(n/2)} k * A224344(n,k). - Alois P. Heinz, Aug 06 2019

More terms from Joshua Zucker, May 10 2006

A336579 Sum of prime parts, counted without multiplicity, in all compositions of n.

Original entry on oeis.org

0, 0, 2, 7, 14, 38, 83, 193, 421, 917, 1969, 4210, 8908, 18763, 39287, 81940, 170270, 352726, 728663, 1501711, 3088326, 6339424, 12991312, 26583389, 54323352, 110876435, 226057023, 460432903, 936963134, 1905110662, 3870698364, 7858803605, 15945759386

Offset: 0

Alois P. Heinz, Jul 26 2020

nonn

			a(4) = 2 + 2 + 2 + 2 + 3 + 3 = 14: 1111, 11(2), 1(2)1, (2)11, (2)2, 1(3), (3)1, 4.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000040, A097939, A102712, A309561, A336516, A336632.

b:= proc(n, i, p) option remember; `if`(n=0, [p!, 0],
      `if`(i<1, 0, add((p-> [0, `if`(j>0 and isprime(i),
       p[1]*i, 0)]+p)(b(n-i*j, i-1, p+j)/j!), j=0..n/i)))
    end:
a:= n-> b(n$2, 0)[2]:
seq(a(n), n=0..38);

b[n_, i_, p_] := b[n, i, p] = If[n == 0, {p!, 0},
     If[i < 1, {0, 0}, Sum[Function[q, {0, If[j > 0 && PrimeQ[i],
     q[[1]]*i, 0]} + q][b[n - i*j, i - 1, p + j]/j!], {j, 0, n/i}]]];
a[n_] := b[n, n, 0][[2]];
Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Mar 17 2022, after Alois P Heinz *)

cp's OEIS Frontend

A102291 Total number of prime parts in all compositions of n.

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