A002121 a(0) = 1, a(1) = 0, a(2) = -1; for n >= 3, a(n) = - a(n-2) + Sum_{ primes p with 3 <= p <= n} a(n-p). (Formerly M0023 N0005)

1, 0, -1, 1, 1, -1, 0, 2, 0, -2, 2, 4, -3, -2, 8, 1, -8, 8, 12, -11, -4, 25, 4, -24, 21, 40, -31, -16, 82, 14, -81, 71, 131, -99, -48, 258, 46, -249, 223, 422, -303, -162, 825, 169, -791, 714, 1360, -955, -503, 2641, 573, -2479, 2263, 4365, -2941, -1592, 8436, 1978, -7830, 7212, 14083, -9133, -4992, 26970, 6688, -24590

Offset

0,8

Comments

Arises in studying the Goldbach conjecture.

The last negative term appears to be a(303). - T. D. Noe, Dec 05 2006

References

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

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

Links

T. D. Noe, Table of n, a(n) for n = 0..1000

P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380. [The sequence g_n]

Index entries for sequences related to Goldbach conjecture

Formula

G.f.: 1/(1+Sum_{k>0} (-x)^prime(k)). - Vladeta Jovovic, Mar 29 2003

Mathematica

CoefficientList[Series[1/(1+Sum[(-x)^Prime[k], {k, 20}]), {x, 0, 70}], x] (* Harvey P. Dale, Aug 26 2020 *)

Prog

(Haskell)
import Data.List (genericIndex)
a002121 n = genericIndex a002121_list n
a002121_list = 1 : 0 : -1 : f 0 (-1) 3 where
   f v w x = y : f w y (x + 1) where
     y = sum (map (a002121 . (x -)) $ takeWhile (<= x) a065091_list) - v
-- Reinhard Zumkeller, Mar 21 2014

Crossrefs

Cf. A002100-A002125.

Cf. A065091.

Sequence in context: A181346 A300815 A358933 * A279158 A273166 A331262

Adjacent sequences: A002118 A002119 A002120 * A002122 A002123 A002124

Keyword

sign,easy,look

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Mar 29 2003

Entry revised by N. J. A. Sloane, Dec 04 2006

Status

approved