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

%I A376552 #39 Jul 02 2025 16:54:30
%S A376552 1,24,215,44732416,445902212680990209,
%T A376552 2470738560300573839567485058051752329216,
%U A376552 194775879942444285383551347529278187374780378665463617801353369255538909241232419740031
%N A376552 Square root of the product of all sums and differences of the square roots of the first n primes.
%C A376552 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.
%H A376552 Lucas A. Brown, <a href="/A376552/b376552.txt">Table of n, a(n) for n = 2..10</a>
%H A376552 Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A354913.py">Python program</a>.
%H A376552 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Swinnerton-DyerPolynomial.html">Swinnerton-Dyer Polynomial</a>.
%H A376552 Wikipedia, <a href="https://en.wikipedia.org/wiki/Swinnerton-Dyer_polynomial">Swinnerton-Dyer polynomial</a>.
%F A376552 a(n) = sqrt(A354913(n)).
%e A376552 The Swinnerton-Dyer polynomial for n=1 is x^2 - 2, which has negative constant term, so we skip n = 1.
%e A376552 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.
%e A376552 For n = 3, the Swinnerton-Dyer polynomial is x^8 - 40*x^6 + 352*x^4 - 960*x^2 + 576, so a(3) = 24.
%p A376552 p:= proc(n) option remember; expand(`if`(n=0, x, mul(
%p A376552       subs(x=x+i*sqrt(ithprime(n)), p(n-1)), i=[1, -1])))
%p A376552     end:
%p A376552 a:= n-> isqrt(coeff(p(n), x, 0)):
%p A376552 seq(a(n), n=2..8);  # _Alois P. Heinz_, Nov 28 2024
%t A376552 p[n_] := p[n] = Expand[If[n == 0, x, Product[p[n - 1] /. x -> x + i*Sqrt[Prime[n]], {i, {1, -1}}]]];
%t A376552 a[n_] := Coefficient[p[n], x, 1 - Sign[n]] // Sqrt // Floor;
%t A376552 Table[a[n], {n, 2, 10}] (* _Jean-François Alcover_, Jul 02 2025, after _Alois P. Heinz_ *)
%o A376552 (Python) # See LINKS.
%Y A376552 Cf. A000040, A153731, A354913.
%K A376552 nonn
%O A376552 2,2
%A A376552 _Lucas A. Brown_, Nov 27 2024