A002098 G.f.: 1/Product_{k>=1} (1-prime(k)*x^prime(k)). (Formerly M0597 N0215)

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

0,3

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).

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 *)

Prog

(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

Crossrefs

Cf. A002099, A006906.

Row sums of A064364, A116864.

Sequence in context: A061919 A328883 A192613 * A301318 A297180 A162969

Adjacent sequences: A002095 A002096 A002097 * A002099 A002100 A002101

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved