A002125 a(n) = Sum_{k=0..n} f(k)*f(n-k) where f(k) = A002124(k). (Formerly M0024 N0006)

1, 0, 0, 2, 0, 2, 3, 2, 6, 4, 9, 14, 11, 26, 29, 34, 62, 68, 99, 140, 169, 252, 322, 430, 607, 764, 1059, 1424, 1845, 2546, 3344, 4442, 6002, 7876, 10575, 14058, 18575, 24878, 32842, 43630, 58073, 76658, 101913, 134964, 178468, 236776, 312874, 414094, 547947, 723646

Offset

0,4

Comments

Arises in studying the Goldbach conjecture.

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. 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 I_n]

Formula

G.f.: 1/(1 - Sum_{k>=2} x^prime(k))^2. - Ilya Gutkovskiy, Apr 11 2017

Maple

M:=120; f:=array(0..M); f[0]:=1; f[1]:=0; f[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 + f[n-p]; fi; od: f[n]:=t1; od: # f is A002124
A002125:=array(0..M); for n from 0 to M do A002125[n]:=add(f[t]*f[n-t], t=0..n); od: [seq(A002125[n], n=0..M)];

Mathematica

CoefficientList[Series[1/(1 - Sum[x^Prime[k], {k, 2, 50}])^2, {x, 0, 50}], x] (* Indranil Ghosh, Apr 12 2017 *)

Prog

(Haskell)
a002125 n = a002125_list !! n
a002125_list = uncurry conv $ splitAt 1 a002124_list where
   conv xs (z:zs) = sum (zipWith (*) xs $ reverse xs) : conv (z:xs) zs
-- Reinhard Zumkeller, Mar 21 2014

Crossrefs

Sequence in context: A033769 A333636 A074660 * A375201 A171731 A323212

Adjacent sequences: A002122 A002123 A002124 * A002126 A002127 A002128

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved