A371757 Numbers of the form (4^(p^(k+1)) + 1)/(4^(p^k) + 1), where k >= 1 and p is an odd prime other than 5.

4033, 68719214593, 19341632594266545643831297, 324518553658426708768757511094273, 1684996264962499703367587717863072443065045481313942556034056847361

Offset: 1

Author

Amiram Eldar, Apr 05 2024

Keywords

• nonn

Comments

Mąkowski and Rotkiewicz (1969) proved that all the terms are Fermat pseudoprimes to base 2 (A001567).

The next term has 94 digits and is too large to include in the data section.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..18
• Andrzej Mąkowski and Andrzej Rotkiewicz, On pseudoprime numbers of special form, Colloquium Mathematicum, Vol. 2, No. 20 (1969), pp. 269-271; alternative link.
• Wikipedia, Pseudoprime.
• Index entries for sequences related to pseudoprimes.

Crossrefs

Subsequence of A001567.

Cf. A210454, A293626.

Programs

Mathematica

f[p_, k_] := (4^(p^(k + 1)) + 1)/(4^(p^k) + 1);
seq[max_] := Module[{s = {}, p = 3, f1, k, addFlag = True}, While[addFlag, If[p == 5, Continue[]]; k = 1; addFlag = False; While[(f1 = f[p, k]) < max, AppendTo[s, f1]; addFlag = True; k++]; p = NextPrime[p, If[p == 3, 2, 1]]]; Sort[s]]; seq[10^80]