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%I #21 Jan 27 2024 20:26:27
%S 1,0,0,0,0,0,2,0,4,0,14,0,38,0,126,0,394,0,1290,0,4344,0,14846,0,
%T 51068,0,178436,0,634568,0,2261052,0,8067296,0,29031484,0,105251904,0,
%U 383580180,0,1404666680,0,5171079172,0,19141098744,0,71125205900,0,263549059326
%N Number of solutions to c(1)*prime(1) + ... + c(n)*prime(n) = 2, where c(i) = +-1 for i > 1, c(1) = 1.
%H Alois P. Heinz, <a href="/A022896/b022896.txt">Table of n, a(n) for n = 1..500</a>
%F 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
%F a(n) = [x^0] Product_{k=2..n} (x^prime(k) + 1/x^prime(k)). - _Ilya Gutkovskiy_, Jan 26 2024
%e a(7) counts these 2 solutions: {2, -3, -5, -7, 11, -13, 17}, {2, 3, 5, 7, -11, 13, -17}.
%t {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}]
%t (* A022896, a(n) = number of solutions of "sum = s" using Prime(f) to Prime(f+n-1) *)
%t 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 *)
%o (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
%o (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
%Y 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
%K nonn
%O 1,7
%A _Clark Kimberling_
%E Corrected and extended by _Clark Kimberling_, Oct 01 2013
%E a(23)-a(49) from _Alois P. Heinz_, Aug 06 2015
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