A046676 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).

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, 69, 78, 89, 99, 113, 126, 143, 159, 179, 199, 224, 248, 277, 307, 343, 378, 421, 464, 515, 567, 628, 690, 763, 837, 923, 1012, 1115, 1219, 1340, 1465, 1607

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

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

References

• 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.

Links

• George E. Andrews, Arnold Knopfmacher, John Knopfmacher, Engel expansions and the Rogers-Ramanujan identities J. Number Theory 80 (2000), 273-290. See Eq. 2.1.

Crossrefs

Differs from A000607 at the 20th term. Cf. A192541.

Programs

Maple

t3:=1+add(q^sum(ithprime(i),i=1..j)/mul(1-q^i,i=1..j), j=1..51);
t4:=series(t3,q,50);
t5:=seriestolist(%);

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

PARI

A046676(n,S=1,P=1+O(x^(n+1)))={for(k=1,n, nM. F. Hasler, Mar 05 2014