A376552 - OEIS

%I #34 Nov 29 2024 21:39:24

%S 1,24,215,44732416,445902212680990209,

%T 2470738560300573839567485058051752329216,

%U 194775879942444285383551347529278187374780378665463617801353369255538909241232419740031

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

%C 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 Lucas A. Brown, <a href="/A376552/b376552.txt">Table of n, a(n) for n = 2..10</a>

%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A354913.py">Python program</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Swinnerton-DyerPolynomial.html">Swinnerton-Dyer Polynomial</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Swinnerton-Dyer_polynomial">Swinnerton-Dyer polynomial</a>.

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

%e The Swinnerton-Dyer polynomial for n=1 is x^2 - 2, which has negative constant term, so we skip n = 1.

%e 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 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 p:= proc(n) option remember; expand(`if`(n=0, x, mul(

%p       subs(x=x+i*sqrt(ithprime(n)), p(n-1)), i=[1, -1])))

%p     end:

%p a:= n-> isqrt(coeff(p(n), x, 0)):

%p seq(a(n), n=2..8);  # _Alois P. Heinz_, Nov 28 2024

%o (Python) # See LINKS.

%Y Cf. A000040, A153731, A354913.

%K nonn

%O 2,2

%A _Lucas A. Brown_, Nov 27 2024