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%I #23 Mar 08 2020 00:24:29
%S 1,0,1,1,1,2,2,3,3,4,5,6,7,9,10,12,14,17,19,23,26,31,35,41,46,54,60,
%T 69,78,89,99,113,126,143,159,179,199,224,248,277,307,343,378,421,464,
%U 515,567,628,690,763,837,923,1012,1115,1219,1340,1465,1607
%N Expansion of 1 + Sum_{k>=1} x^(p_1+p_2+...+p_k)/((1-x)*(1-x^2)*(1-x^3)*...*(1-x^k)) (where p_i are the primes).
%C Ramanujan considered that this could equal the prime parts partition numbers A000607, but they differ from the 20th term on, cf. A192541. See A238804 for a correct variant, where the coefficient and power of x^{...} are adjusted to match A000607. - _M. F. Hasler_, Mar 06 2014
%D B. C. Berndt and B. M. Wilson, Chapter 5 of Ramanujan's second notebook, pp. 49-78 of Analytic Number Theory (Philadelphia, 1980), Lect. Notes Math. 899, 1981, see Entry 29.
%H George E. Andrews, Arnold Knopfmacher, John Knopfmacher, <a href="http://dx.doi.org/10.1006/jnth.1999.2453">Engel expansions and the Rogers-Ramanujan identities</a> J. Number Theory 80 (2000), 273-290. See Eq. 2.1.
%p t3:=1+add(q^sum(ithprime(i),i=1..j)/mul(1-q^i,i=1..j), j=1..51);
%p t4:=series(t3,q,50);
%p t5:=seriestolist(%);
%o (PARI) Vec(sum(i=0,25,x^sum(k=1,i,prime(k))/prod(k=1,i,1-x^k),O(x^99))) \\ _M. F. Hasler_, Mar 05 2014
%o (PARI) A046676(n,S=1,P=1+O(x^(n+1)))={for(k=1,n, n<valuation(P*=x^prime(k)/(1-x^k),x)&&break;S+=P); polcoeff(S,n)} \\ _M. F. Hasler_, Mar 05 2014
%Y Differs from A000607 at the 20th term. Cf. A192541.
%K nonn
%O 0,6
%A _N. J. A. Sloane_
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