A002124 Number of compositions of n into a sum of odd primes. (Formerly M0154 N0062)

1, 0, 0, 1, 0, 1, 1, 1, 2, 1, 3, 4, 3, 7, 7, 8, 14, 15, 21, 28, 33, 47, 58, 76, 103, 125, 169, 220, 277, 373, 476, 616, 810, 1037, 1361, 1763, 2279, 2984, 3846, 5006, 6521, 8428, 10983, 14249, 18480, 24048, 31178, 40520, 52635, 68281, 88765, 115211, 149593, 194381, 252280, 327696, 425587, 552527, 717721

Offset

0,9

Comments

Arises in studying the Goldbach conjecture.

The g.f. -(z-1)*(z+1)*(z**2+z+1)*(z**2-z+1)/(1-z**6-z**3-z**5-z**7+z**9) conjectured by Simon Plouffe in his 1992 dissertation is wrong.

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

N. J. A. Sloane, Table of n, a(n) for n = 0..1000

P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; see page 300

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. 354-382] [The sequence i_n]

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.

Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Index entries for sequences related to Goldbach conjecture

Formula

a(0)=1, a(1)=a(2)=0; for n >= 3, a(n) = Sum_{ primes p with 3 <= p <= n} a(n-p). [MacMahon]

G.f.: 1/( 1 - Sum_{k>=2} x^A000040(k) ). [Joerg Arndt, Sep 30 2012]

Maple

A002124 := proc(n) coeff(series(1/(1-add(z^numtheory[ithprime](j), j=2..n)), z=0, n+1), z, n) end;
M:=120; a:=array(0..M); a[0]:=1; a[1]:=0; a[2]:=0; for n from 3 to M do t1:=0; for k from 2 to n do p := ithprime(k); if p <= n then t1 := t1 + a[n-p]; fi; od: a[n]:=t1; od: [seq(a[n], n=0..M)]; # N. J. A. Sloane, after MacMahon, Dec 03 2006; used in A002125

Mathematica

a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = (s = 0; p = 3; While[p <= n, s = s + a[n-p]; p = NextPrime[p]]; s); a /@ Range[0, 58] (* Jean-François Alcover, Jun 28 2011, after P. A. MacMahon *)

Prog

(Haskell)
import Data.List (genericIndex)
a002124 n = genericIndex a002124_list n
a002124_list = 1 : f 1 [] a065091_list where
   f x qs ps'@(p:ps)
     | p <= x    = f x (p:qs) ps
     | otherwise = sum (map (a002124 . (x -)) qs) : f (x + 1) qs ps'
-- Reinhard Zumkeller, Mar 21 2014

Crossrefs

Cf. A002125, A023360, A024939, A077608.

Cf. A065091.

Sequence in context: A133310 A362563 A077608 * A097564 A345233 A128270

Adjacent sequences: A002121 A002122 A002123 * A002125 A002126 A002127

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Better description and more terms from Philippe Flajolet, Nov 11 2002

Edited by N. J. A. Sloane, Dec 03 2006

Status

approved