A261059 Number of solutions to c(1)*prime(2)+...+c(2n)*prime(2n+1) = -2, where c(i) = +-1 for i > 1, c(1) = 1.

1, 0, 2, 1, 4, 25, 47, 237, 562, 1965, 7960, 24148, 85579, 307569, 1104519, 4106381, 14710760, 52113647, 193181449, 698356631, 2574590311, 9600573372, 35644252223, 131545038705, 492346772797, 1843993274342, 6903884199622, 25984680496124, 97937400336407

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 all terms are odd but the r.h.s. is even.

Examples

			a(1) = 1 because prime(2) - prime(3) = -2.
a(2) = 0 because prime(2) +- prime(3) +- prime(4) +- prime(5) is different from -2 for any choice of the signs.
a(3) = 2 counts the 2 solutions prime(2) - prime(3) + prime(4) - prime(5) - prime(6) + prime(7) = -2 and prime(2) - prime(3) - prime(4) + prime(5) + prime(6) - prime(7) = -2.
		

Links

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

Crossrefs

Cf. A261057 (starting with prime(1)), A261060 (starting with prime(3)), A261045 (starting with prime(4)), A261061 - A261063 and A261044 (r.h.s. = -1), A022894 - A022904, A083309, A022920 (r.h.s. = 0, 1 or 2), .

Programs

Maple

s:= proc(n) option remember;
      `if`(n<3, 0, ithprime(n)+s(n-1))
    end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=2, 1,
      b(abs(n-ithprime(i)),i-1)+b(n+ithprime(i),i-1)))
    end:
a:= n-> b(5, 2*n+1):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 08 2015

Mathematica

s[n_] := s[n] = If[n<3, 0, Prime[n]+s[n-1]]; b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 2, 1, b[Abs[n-Prime[i]], i-1] + b[n+Prime[i], i-1]]];  a[n_] := b[5, 2*n+1]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

PARI

A261059(n,rhs=-2,firstprime=2)={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.

PARI

a(n,s=-2-3,p=2)=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,a(abs(s),sum(i=p+1,p+2*n-1,prime(i)),prime(p+n*2-1))))

Formula

a(n) = [x^5] Product_{k=3..2*n+1} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 31 2024

Extensions

a(15)-a(29) from Alois P. Heinz, Aug 08 2015