A292699 Number of solutions to 5 +- 11 +- 17 +- 23 +- ... +- prime(4*n+1) = 0.

0, 1, 1, 2, 2, 5, 28, 102, 242, 835, 3381, 10115, 39415, 138555, 465778, 1741167, 6081563, 22156403, 81090107, 301185788, 1098440558, 4071519963, 15235221189, 56764430821, 211797564907, 790029575790, 2962705350767, 11154931819979, 42097079364709

Offset

1,4

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) + 1/x^prime(2*k+1)).

Example

For n=2 the solution is 5-11-17+23 = 0.
For n=3 the solution is 5+11+17-23+31-41 = 0.
For n=4 the 2 solutions are 5-11+17+23+31+41-47-59 = 0 and 5+11-17+23+31-41+47-59 = 0.

Maple

s:= proc(n) s(n):= `if`(n=0, 0, ithprime(2*n+1)+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+1))))
    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+1] + s[n-1]];
b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 0, 1,
     With[{p = Prime[2i+1]}, 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)+1/x^prime(2*k+1)), 0)}

Crossrefs

Cf. A000040, A022894, A083309, A292698, A292700.

Sequence in context: A384274 A154647 A103890 * A014566 A259861 A293264

Adjacent sequences: A292696 A292697 A292698 * A292700 A292701 A292702

Keyword

nonn

Author

Seiichi Manyama, Sep 21 2017

Status

approved