A046675 Expansion of Product_{i>0} (1-x^{p_i}), where p_i are the primes.

1, 0, -1, -1, 0, 0, 0, 0, 1, 1, 0, -1, 0, 0, 0, 0, 1, 0, 0, -1, 0, 0, 0, -1, 1, 1, 0, -1, 0, -1, 0, -1, 1, 1, 1, -1, 1, -1, -1, -1, 2, 0, 1, -1, 1, 0, 0, -3, 2, 1, 1, -2, 1, -2, 1, -2, 1, 0, 2, -3, 3, -1, 0, -2, 4, -1, 2, -4, 1, -1, 3, -5, 4, -1, 2, -3, 4, -4, 3, -5, 3, -1, 4, -8, 6, -1, 2, -7, 6, -4, 8, -6, 3

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

Comments

The difference between the number of even partitions of n into distinct primes and the number of odd partitions of n into distinct primes. - T. D. Noe, Sep 08 2006

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
• H. Gupta, Partitions into distinct primes, Proc. Nat. Acad. Sci. India, 21 (1955), 185-187 [broken link].

Crossrefs

Cf. A000607, A000586.

Programs

Mathematica

CoefficientList[Series[Product[1 - x^Prime[i], {i, 1, 100}], {x, 0, 100}], x] (* Vaclav Kotesovec, Sep 13 2018 *)
nmax = 100; pmax = PrimePi[nmax]; poly = ConstantArray[0, nmax + 1]; poly[[1]] = 1; poly[[2]] = 0; poly[[3]] = -1; Do[p = Prime[k]; Do[poly[[j]] -= poly[[j - p]], {j, nmax + 1, p + 1, -1}];, {k, 2, pmax}]; poly (* Vaclav Kotesovec, Sep 20 2018 *)

Formula

a(n) = A184171(n) - A184172(n). - R. J. Mathar, Jan 10 2011

Extensions

Revised by N. J. A. Sloane, Jun 10 2012