A022896 Number of solutions to c(1)*prime(1) + ... + c(n)*prime(n) = 2, where c(i) = +-1 for i > 1, c(1) = 1.

1, 0, 0, 0, 0, 0, 2, 0, 4, 0, 14, 0, 38, 0, 126, 0, 394, 0, 1290, 0, 4344, 0, 14846, 0, 51068, 0, 178436, 0, 634568, 0, 2261052, 0, 8067296, 0, 29031484, 0, 105251904, 0, 383580180, 0, 1404666680, 0, 5171079172, 0, 19141098744, 0, 71125205900, 0, 263549059326

Offset

1,7

Links

Alois P. Heinz, Table of n, a(n) for n = 1..500

Formula

a(2n-1) = A113041(n) - A261057(n), a(2n) = 0 because there is an odd number of odd terms on the left hand side, but the right hand side is even. - M. F. Hasler, Aug 09 2015

a(n) = [x^0] Product_{k=2..n} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 26 2024

Example

a(7) counts these 2 solutions: {2, -3, -5, -7, 11, -13, 17}, {2, 3, 5, 7, -11, 13, -17}.

Mathematica

{f, s} = {1, 2}; Table[t = Map[Prime[# + f - 1] &, Range[2, z]]; Count[Map[Apply[Plus, #] &, Map[t # &, Tuples[{-1, 1}, Length[t]]]], s - Prime[f]], {z, 22}]
(* A022896, a(n) = number of solutions of "sum = s" using Prime(f) to Prime(f+n-1) *)
n = 7; t = Map[Prime[# + f - 1] &, Range[n]]; Map[#[[2]] &, Select[Map[{Apply[Plus, #], #} &, Map[t # &, Map[Prepend[#, 1] &, Tuples[{-1, 1}, Length[t] - 1]]]], #[[1]] == s &]]  (* the 2 solutions of using n=7 primes; Peter J. C. Moses, Oct 01 2013 *)

Prog

(PARI) A022896(n, rhs=2, firstprime=1)={rhs-=prime(firstprime); my(p=vector(n-1, i, prime(i+firstprime))); !(rhs||#p)+sum(i=1, 2^#p-1, sum(j=1, #p, (-1)^bittest(i, j-1)*p[j])==rhs)} \\ For illustrative purpose, too slow for n >> 20. - M. F. Hasler, Aug 08 2015
(PARI) a(n, s=2-prime(1), p=1)={if(n<=s, if(s==p, n==s, a(abs(n-p), s-p, precprime(p-1))+a(n+p, s-p, precprime(p-1))), if(s<=0, if(n>1, a(abs(s), sum(i=p+1, p+n-1, prime(i)), prime(p+n-1)), !s)))} \\ M. F. Hasler, Aug 09 2015

Crossrefs

Cf. A022894 (r.h.s. = 0), A022895 (r.h.s. = 1), A022897, ..., A022904, A022920 (using primes >= 7), A083309; A261061 - A261063 and A261045 (r.h.s. = -1); A261057, A261059, A261060 and A261044 (r.h.s. = -2); A113040, A113041, A113042. - M. F. Hasler, Aug 08 2015

Sequence in context: A286122 A286776 A265829 * A100225 A007420 A362368

Adjacent sequences: A022893 A022894 A022895 * A022897 A022898 A022899

Keyword

nonn

Author

Clark Kimberling

Extensions

Corrected and extended by Clark Kimberling, Oct 01 2013

a(23)-a(49) from Alois P. Heinz, Aug 06 2015

Status

approved