A113040 Number of solutions to +-p(1)+-p(2)+-...+-p(2n)=1 where p(i) is the i-th prime.

1, 1, 3, 6, 16, 45, 138, 439, 1417, 4698, 16021, 55146, 190274, 671224, 2404289, 8535117, 30635869, 110496946, 401422210, 1467402238, 5393176633, 19883249002, 73856531314, 273602448261, 1017563027699, 3803902663467, 14266523388813, 53564969402478

Offset: 1

Author

Floor van Lamoen, Oct 12 2005

Keywords

• nonn

Comments

+-p(1)+-p(2)+-...+-p(2n+1)=1 has no solutions because the l.h.s. is even.

Examples

			2 + 3 + 5 - 7 + 11 - 13 = - 2 + 3 + 5 - 7 - 11 + 13 = - 2 + 3 - 5 + 7 + 11 - 13 = 1 so a(3) = 3.
		

Links

• Alois P. Heinz and Ray Chandler, Table of n, a(n) for n = 1..1000 (first 130 terms from Alois P. Heinz)

Crossrefs

Cf. A083309, A022894 - A022904, A022920, A261057 - A261063 and A261044 - A261045; A215036.

Bisection (even part) of A306443.

Programs

Maple

A113040:=proc(n) local i,j,p,t; t:= NULL; for j from 2 to 2*n by 2 do p:=1; for i to j do p:=p*(x^(-ithprime(i))+x^(ithprime(i))); od; t:=t,coeff(p,x,1); od; t; end;
# second Maple program:
sp:= proc(n) sp(n):= `if`(n=0, 0, ithprime(n)+sp(n-1)) end:
b := proc(n,i) option remember; `if`(n>sp(i), 0, `if`(i=0, 1,
        b(n+ithprime(i), i-1)+ b(abs(n-ithprime(i)), i-1)))
     end:
a:= n-> b(1, 2*n):
seq(a(n), n=1..40);  # Alois P. Heinz, Aug 05 2012

Mathematica

sp[n_] := If[n == 0, 0, Prime[n]+sp[n-1]]; b[n_, i_] := b[n, i] =If[n > sp[i], 0, If[i == 0, 1, b[n+Prime[i], i-1] + b[Abs[n-Prime[i]], i-1]]]; a[n_] := b[1, 2*n]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

Formula

a(n) = A022895(2n) + A261061(n). - M. F. Hasler, Aug 09 2015

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

a(n) = [x^1] Product_{k=1..2*n} (x^prime(k) + 1/x^prime(k)). - Ilya Gutkovskiy, Jan 25 2024