A261061 Number of solutions to c(1)*prime(1)+...+c(2n)*prime(2n) = -1, where c(i) = +-1 for i > 1, c(1) = 1.

1, 0, 2, 3, 8, 23, 68, 221, 709, 2344, 8006, 27585, 95114, 335645, 1202053, 4267640, 15317698, 55248527, 200711160, 733697248, 2696576651, 9941588060, 36928160817, 136800727634, 508780005068, 1901946851732, 7133247301621, 26782446410398, 100862459737318

Offset: 1

Author

M. F. Hasler, Aug 08 2015

Keywords

• nonn

Comments

There cannot be a solution for an odd number of terms on the l.h.s. because there would be an even number of odd terms but the r.h.s. is odd.

Examples

			a(1) = 1 counts the solution prime(1) - prime(2) = -1.
a(2) = 0 because prime(1) +- prime(2) +- prime(3) +- prime(4) is always different from -1.
a(3) = 2 counts the two solutions prime(1) - prime(2) + prime(3) - prime(4) - prime(5) + prime(6) = -1 and prime(1) - prime(2) - prime(3) + prime(4) + prime(5) - prime(6) = -1.
		

Links

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

Crossrefs

Cf. A261062 - A261063 and A261044 (starting with prime(2), prime(3) resp. 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<2, 0, ithprime(n)+s(n-1))
    end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=1, 1,
      b(abs(n-ithprime(i)), i-1)+b(n+ithprime(i), i-1)))
    end:
a:= n-> b(3, 2*n):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 08 2015

Mathematica

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

PARI

A261061(n,rhs=-1,firstprime=1)={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

Conjecture: limit_{n->infinity} a(n)^(1/n) = 4. - Vaclav Kotesovec, Jun 05 2019

a(n) = [x^3] Product_{k=2..2*n} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 31 2024

Extensions

a(14)-a(29) from Alois P. Heinz, Aug 08 2015