A002098 G.f.: 1/Product_{k>=1} (1-prime(k)*x^prime(k)).

1, 0, 2, 3, 4, 11, 17, 29, 49, 85, 144, 226, 404, 603, 1025, 1679, 2558, 4201, 6677, 10190, 16599, 25681, 39643, 61830, 96771, 147114, 228338, 352725, 533291, 818624, 1263259, 1885918, 2900270, 4396577, 6595481, 10040029, 15166064, 22642064

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

a(n) is sum of all numbers k for which A001414(k), the sum of prime factors with repetition, equals n. See Havermann's link. - J. M. Bergot, Jun 14 2013

References

• S.M. Kerawala, On a Pair of Arithmetic Functions Analogous to Chawla's Pair, J. Natural Sciences and Mathematics, 9 (1969), circa p. 103.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..6274 (terms 0..500 from T. D. Noe)
• H. Havermann: Tables of sum-of-prime-factors sequences (overview with links to the first 50000 sums).

Crossrefs

Cf. A002099, A006906.

Row sums of A064364, A116864.

Programs

Maple

b:= proc(n,i) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=0 then 0
    else b(n, i-1) +b(n-ithprime(i), i) *ithprime(i)
      fi
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..40);  # Alois P. Heinz, Nov 20 2010

Mathematica

b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i==0, 0, True, b[n, i-1] + b[n - Prime[i], i]*Prime[i]]; a[n_] := b[n, PrimePi[n]]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Feb 19 2016, after Alois P. Heinz *)
With[{nn=40},CoefficientList[Series[1/Product[1-Prime[k]x^Prime[k],{k,nn}],{x,0,nn}],x]] (* Harvey P. Dale, Jun 20 2021 *)

PARI

my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-isprime(k)*k*x^k)) \\ Seiichi Manyama, Feb 27 2022

Extensions

Better description and more terms from Vladeta Jovovic, May 09 2003