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

0,6

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

Table of n, a(n) for n=0..57.

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.

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(%);

Prog

(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, n<valuation(P*=x^prime(k)/(1-x^k), x)&&break; S+=P); polcoeff(S, n)} \\ M. F. Hasler, Mar 05 2014

Crossrefs

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

Sequence in context: A112021 A000607 A114372 * A003114 A185227 A217569

Adjacent sequences: A046673 A046674 A046675 * A046677 A046678 A046679

Keyword

nonn

Author

N. J. A. Sloane

Status

approved