A002123 a(1) = 0, a(2) = 0; for n > 2, a(n) - a(n-3) - a(n-5) - ... - a(n-p) = n if n is prime, otherwise = 0, where p = largest prime < n.

0, 0, 3, 0, 5, -3, 7, -8, 3, -15, 22, -15, 39, -35, 38, -72, 85, -111, 152, -175, 241, -308, 414, -551, 655, -897, 1164, -1463, 2001, -2538, 3286, -4296, 5503, -7259, 9357, -12147, 15910, -20406, 26640, -34703, 44854, -58481, 75809, -98340

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign

Comments

Arises in studying the Goldbach conjecture.

References

• P. A. MacMahon, Properties of prime numbers deduced from the calculus of symmetric functions, Proc. London Math. Soc., 23 (1923), 290-316. [Coll. Papers, Vol. II, pp. 354-382] [The sequence f_n]
• 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 = 1..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.
• Index entries for sequences related to Goldbach conjecture

Crossrefs

Cf. A065091, A061397.

Programs

Haskell

import Data.List (genericIndex)
a002123 n = genericIndex a002123_list (n - 1)
a002123_list = 0 : 0 : f 3 where
   f x = y : f (x + 1) where
     y = a061397 x -
         sum (map (a002123 . (x -)) $ takeWhile (< x) a065091_list)
-- Reinhard Zumkeller, Mar 21 2014

Extensions

Extended with signs by T. D. Noe, Dec 05 2006