A367228 - OEIS

%I #10 Nov 11 2023 10:39:49

%S 15,85,4369,16843009,281479271743489,79228162532711081671548469249,

%T 6277101735386680764176071790128604879584176795969512275969

%N Products of two consecutive Fermat numbers: a(n) = A000215(n) * A000215(n+1).

%C a(7) has 116 digits and is too large to include in the data section.

%C Szymiczek (1966) proved that a(n) is a super-Poulet number (A050217) for all n >= 2. All the composite Fermat numbers (A281576) are also super-Poulet numbers.

%D Michal Krížek, Florian Luca and Lawrence Somer, 17 Lectures on Fermat Numbers, Springer-Verlag, N.Y., 2001, p. 142.

%H Amiram Eldar, <a href="/A367228/b367228.txt">Table of n, a(n) for n = 0..10</a>

%H Andrzej Rotkiewicz, <a href="https://eudml.org/doc/35153">On pseudoprimes having special forms and a solution of K. Szymiczek's problem</a>, Acta Mathematica Universitatis Ostraviensis, Vol. 13, No. 1 (2005), pp. 57-71.

%H Kazimierz Szymiczek, <a href="https://www.e-periodica.ch/digbib/view?pid=edm-001%3A1966%3A21%3A%3A4&amp;referrer=search#65">Note on Fermat numbers</a>, Elemente der Mathematik, Vol. 21, No. 3 (1966), p. 59.

%F a(n) = (2^(2^n) + 1) * (2^(2^(n+1)) + 1).

%t f[n_] := 2^(2^n) + 1; a[n_] := f[n] * f[n + 1]; Array[a, 7, 0]

%o (PARI) f(n) = 2^(2^n) + 1;

%o a(n) = f(n) * f(n+1);

%Y Cf. A000215, A050217, A281576.

%K nonn,easy

%O 0,1

%A _Amiram Eldar_, Nov 11 2023