A350880 a(n) is the constant term in expansion of Product_{k=1..n} (x^prime(k) + 1 + 1/x^prime(k)).

1, 1, 1, 3, 5, 7, 17, 39, 95, 233, 561, 1435, 3643, 9417, 24973, 66695, 177915, 475629, 1293017, 3517223, 9636365, 26676197, 73848517, 205382439, 571628347, 1588203787, 4435819313, 12474619295, 35194448271, 99782519701, 283514955585, 799783925547

Offset: 0

Author

Ilya Gutkovskiy, Jan 20 2022

Keywords

• nonn

Comments

a(n) is the number of solutions to 0 = Sum_{i=1..n} c_i * prime(i) with c_i in {-1,0,1}. a(3) = 3: -2-3+5, +2+3-5, 0+0+0. - Alois P. Heinz, Dec 28 2023

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400

Crossrefs

Cf. A000040, A007576, A022894.

Programs

Maple

s:= proc(n) s(n):= `if`(n<1, 0, ithprime(n)+s(n-1)) end:
b:= proc(n, i) option remember; `if`(n>s(i), 0, `if`(i=0, 1,
      b(n, i-1)+b(n+ithprime(i), i-1)+b(abs(n-ithprime(i)), i-1)))
    end:
a:= n-> b(0, n):
seq(a(n), n=0..40);  # Alois P. Heinz, Dec 28 2023

Mathematica

s[n_] := s[n] = If[n < 1, 0, Prime[n] + s[n-1]];
b[n_, i_] := b[n, i] = If[n > s[i], 0, If[i == 0, 1, b[n, i-1] + b[n + Prime[i], i-1] + b[Abs[n - Prime[i]], i-1]]];
a[n_] := b[0, n];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Apr 27 2025, after Alois P. Heinz *)

PARI

a(n) = polcoef (prod(k=1, n, x^prime(k) + 1 + 1/x^prime(k)), 0); \\ Michel Marcus, Jan 21 2022