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A022900 Number of solutions to c(1)*prime(3) + ... + c(n)*prime(n+2) = 0, where c(i) = +-1 for i>1, c(1) = 1.

%I A022900 #19 Jan 29 2024 10:52:03
%S A022900 0,0,0,1,0,1,0,1,0,8,0,22,0,42,0,147,0,663,0,1803,0,7410,0,22463,0,
%T A022900 87397,0,291211,0,1091736,0,3896012,0,13992225,0,49681944,0,184771042,
%U A022900 0,677854904,0,2495656379,0,9260633829,0,34281074654,0,127420198855,0
%N A022900 Number of solutions to c(1)*prime(3) + ... + c(n)*prime(n+2) = 0, where c(i) = +-1 for i>1, c(1) = 1.
%H A022900 Alois P. Heinz, <a href="/A022900/b022900.txt">Table of n, a(n) for n = 1..500</a>
%F A022900 a(n) = [x^5] Product_{k=4..n+2} (x^prime(k) + 1/x^prime(k)). - _Ilya Gutkovskiy_, Jan 28 2024
%e A022900 a(8) counts the unique solution {5, -7, 11, -13, 17, -19, -23, 29}.
%t A022900 {f, s} = {3, 0}; 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 A022900 (* A022900, a(n) = number of solutions of "sum = s" using Prime(f) to Prime(f+n-1) *)
%t A022900 n = 8; 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 unique solution of using n=8 primes; _Peter J. C. Moses_, Oct 01 2013 *)
%o A022900 (PARI) A022900(n,rhs=0,firstprime=3)={rhs-=prime(firstprime);my(p=vector(n-1,i,prime(i+firstprime)));sum(i=1,2^(n-1),sum(j=1,#p,(1-bittest(i,j-1)<<1)*p[j])==rhs)} \\ For illustrative purpose, too slow for n >> 20. - _M. F. Hasler_, Aug 08 2015
%Y A022900 Cf. A022894 - A022904, A022920.
%Y A022900 Cf. A261061 - A261063 and A261045 (r.h.s. = -1); A261057, A261059, A261060, A261045 (r.h.s. = -2).
%K A022900 nonn
%O A022900 1,10
%A A022900 _Clark Kimberling_
%E A022900 Corrected and extended by _Clark Kimberling_, Oct 01 2013
%E A022900 a(23)-a(49) from _Alois P. Heinz_, Aug 06 2015
%E A022900 Missing cross-references added by _M. F. Hasler_, Aug 08 2015