A376552 Square root of the product of all sums and differences of the square roots of the first n primes.

1, 24, 215, 44732416, 445902212680990209, 2470738560300573839567485058051752329216, 194775879942444285383551347529278187374780378665463617801353369255538909241232419740031

Offset: 2

Author

Lucas A. Brown, Nov 27 2024

Keywords

• nonn

Comments

a(n) is the square root of the constant term of the Swinnerton-Dyer polynomial for the set {2, 3, 5, ..., prime(n)}. The constant terms themselves are A354913(n) for n >= 1; the nonzero coefficients of the polynomials are A153731.

Examples

			The Swinnerton-Dyer polynomial for n=1 is x^2 - 2, which has negative constant term, so we skip n = 1.
For n = 2, the Swinnerton-Dyer polynomial is (x + sqrt(2) + sqrt(3)) * (x + sqrt(2) - sqrt(3)) * (x - sqrt(2) + sqrt(3)) * (x - sqrt(2) - sqrt(3)) = x^4 - 10*x^2 + 1, so a(2) = 1.
For n = 3, the Swinnerton-Dyer polynomial is x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576, so a(3) = 24.
		

Links

• Lucas A. Brown, Table of n, a(n) for n = 2..10
• Lucas A. Brown, Python program.
• Eric Weisstein's World of Mathematics, Swinnerton-Dyer Polynomial.
• Wikipedia, Swinnerton-Dyer polynomial.

Crossrefs

Cf. A000040, A153731, A354913.

Programs

Maple

p:= proc(n) option remember; expand(`if`(n=0, x, mul(
      subs(x=x+i*sqrt(ithprime(n)), p(n-1)), i=[1, -1])))
    end:
a:= n-> isqrt(coeff(p(n), x, 0)):
seq(a(n), n=2..8);  # Alois P. Heinz, Nov 28 2024

Mathematica

p[n_] := p[n] = Expand[If[n == 0, x, Product[p[n - 1] /. x -> x + i*Sqrt[Prime[n]], {i, {1, -1}}]]];
a[n_] := Coefficient[p[n], x, 1 - Sign[n]] // Sqrt // Floor;
Table[a[n], {n, 2, 10}] (* Jean-François Alcover, Jul 02 2025, after Alois P. Heinz *)

Python

# See LINKS.

Formula

a(n) = sqrt(A354913(n)).