A261063 Number of solutions to c(1)*prime(3) + ... + c(2n-1)*prime(2n+1) = -1, where c(i) = +-1 for i > 1, c(1) = 1.

0, 0, 0, 1, 6, 8, 40, 67, 373, 1232, 3330, 13656, 47111, 164957, 582042, 1967152, 7129046, 26655235, 94956602, 353789267, 1300061367, 4765080122, 17726643505, 66038899483, 245431428625, 919911458949, 3457983108462, 12974054097333, 49016641868213, 185510228030858

Offset: 1

Author

M. F. Hasler, Aug 08 2015

Keywords

• nonn

Comments

There cannot be a solution for an even number of terms on the l.h.s. because all terms are odd but the r.h.s. is odd, too.

Examples

			a(1) = a(2) = 0 because prime(3) and prime(3) +- prime(4) +- prime(5) are different from -1 for any choice of the signs.
a(3) = 0 because the same sums prime(3) +- ... +- prime(7) is also always different from -1 for any choice of the signs.
a(4) = 1 because prime(3) - prime(4) - prime(5) - prime(6) - prime(7) + prime(8) + prime(9) = -1 is the only solution.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..300

Crossrefs

Cf. A261061 - A261062 (starting with prime(1) resp. prime(2)), A261044 (starting with prime(4)), A022894 - A022904, A083309, A022920 (r.h.s. = 0, 1 or 2), A261057, A261059, A261060, A261045 (r.h.s. = -2).

Programs

Maple

s:= proc(n) option remember;
      `if`(n<4, 0, ithprime(n)+s(n-1))
    end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=3, 1,
      b(abs(n-ithprime(i)),i-1)+b(n+ithprime(i),i-1)))
    end:
a:= n-> b(6, 2*n+1):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 08 2015

Mathematica

s[n_] := s[n] = If[n<4, 0, Prime[n]+s[n-1]]; b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 3, 1, b[Abs[n-Prime[i]], i-1] + b[n+Prime[i], i-1]]]; a[n_] := b[6, 2*n+1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

PARI

A261063(n,rhs=-1,firstprime=3)={rhs-=prime(firstprime);my(p=vector(2*n-2+bittest(rhs,0),i,prime(i+firstprime)));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 >> 10.

Formula

a(n) = [x^6] Product_{k=4..2*n+1} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 31 2024

Extensions

a(15)-a(30) from Alois P. Heinz, Aug 08 2015