A292698 Number of solutions to 3 +- 7 +- 13 +- 19 +- ... +- prime(4*n) = 0.

0, 0, 0, 1, 2, 8, 27, 83, 292, 944, 3279, 11291, 38992, 138066, 490248, 1757360, 6347321, 22998089, 83780199, 306819363, 1128999790, 4174251748, 15507225620, 57767819903, 215327188611, 803901214851, 3013081897103, 11331883386143, 42737620941612

Offset

1,5

Links

Seiichi Manyama, Table of n, a(n) for n = 1..200

Formula

Constant term in the expansion of 1/2 * Product_{k=1..2*n} (x^prime(2*k) + 1/x^prime(2*k)).

Example

For n = 4 the solution is 3 - 7 - 13 + 19 - 29 + 37 + 43 - 53 = 0.

Maple

s:= proc(n) s(n):= `if`(n=0, 0, ithprime(2*n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      (p-> b(n+p, i-1)+b(abs(n-p), i-1))(ithprime(2*i))))
    end:
a:= n-> b(0, 2*n)/2:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 21 2017

Mathematica

s[n_] := s[n] = If[n == 0, 0, Prime[2n] + s[n-1]];
b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 0, 1,
     With[{p = Prime[2i]}, b[n+p, i-1] + b[Abs[n-p], i-1]]]];
a[n_] := b[0, 2n]/2;
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Apr 30 2022, after Alois P. Heinz *)

Prog

(PARI) {a(n) = 1/2*polcoeff(prod(k=1, 2*n, x^prime(2*k)+1/x^prime(2*k)), 0)}

Crossrefs

Cf. A000040, A022894, A083309, A292699, A292700.

Sequence in context: A290056 A100505 A102759 * A261056 A295135 A076884

Adjacent sequences: A292695 A292696 A292697 * A292699 A292700 A292701

Keyword

nonn

Author

Seiichi Manyama, Sep 21 2017

Status

approved