A367228 Products of two consecutive Fermat numbers: a(n) = A000215(n) * A000215(n+1).

15, 85, 4369, 16843009, 281479271743489, 79228162532711081671548469249, 6277101735386680764176071790128604879584176795969512275969

Offset: 0

Author

Amiram Eldar, Nov 11 2023

Keywords

• nonn
• easy

Comments

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

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.

References

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

Links

• Amiram Eldar, Table of n, a(n) for n = 0..10
• Andrzej Rotkiewicz, On pseudoprimes having special forms and a solution of K. Szymiczek's problem, Acta Mathematica Universitatis Ostraviensis, Vol. 13, No. 1 (2005), pp. 57-71.
• Kazimierz Szymiczek, Note on Fermat numbers, Elemente der Mathematik, Vol. 21, No. 3 (1966), p. 59.

Crossrefs

Cf. A000215, A050217, A281576.

Programs

Mathematica

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

PARI

f(n) = 2^(2^n) + 1;
a(n) = f(n) * f(n+1);

Formula

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