A002098 - OEIS

%I M0597 N0215 #44 Feb 27 2022 10:23:11

%S 1,0,2,3,4,11,17,29,49,85,144,226,404,603,1025,1679,2558,4201,6677,

%T 10190,16599,25681,39643,61830,96771,147114,228338,352725,533291,

%U 818624,1263259,1885918,2900270,4396577,6595481,10040029,15166064,22642064

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

%C 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

%D S.M. Kerawala, On a Pair of Arithmetic Functions Analogous to Chawla's Pair, J. Natural Sciences and Mathematics, 9 (1969), circa p. 103.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Seiichi Manyama, <a href="/A002098/b002098.txt">Table of n, a(n) for n = 0..6274</a> (terms 0..500 from T. D. Noe)

%H H. Havermann: <a href="http://chesswanks.com/seq/sopfr/">Tables of sum-of-prime-factors sequences (overview with links to the first 50000 sums).</a>

%p b:= proc(n,i) option remember;

%p       if n<0 then 0

%p     elif n=0 then 1

%p     elif i=0 then 0

%p     else b(n, i-1) +b(n-ithprime(i), i) *ithprime(i)

%p       fi

%p     end:

%p a:= n-> b(n, numtheory[pi](n)):

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Nov 20 2010

%t 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_ *)

%t 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 *)

%o (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

%Y Cf. A002099, A006906.

%Y Row sums of A064364, A116864.

%K nonn

%O 0,3

%A _N. J. A. Sloane_

%E Better description and more terms from _Vladeta Jovovic_, May 09 2003