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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.

%I A371757 #12 Jun 23 2024 16:11:56
%S A371757 4033,68719214593,19341632594266545643831297,
%T A371757 324518553658426708768757511094273,
%U A371757 1684996264962499703367587717863072443065045481313942556034056847361
%N 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.
%C A371757 Mąkowski and Rotkiewicz (1969) proved that all the terms are Fermat pseudoprimes to base 2 (A001567).
%C A371757 The next term has 94 digits and is too large to include in the data section.
%H A371757 Amiram Eldar, <a href="/A371757/b371757.txt">Table of n, a(n) for n = 1..18</a>
%H A371757 Andrzej Mąkowski and Andrzej Rotkiewicz, <a href="https://doi.org/10.4064/cm-20-2-269-271">On pseudoprime numbers of special form</a>, Colloquium Mathematicum, Vol. 2, No. 20 (1969), pp. 269-271; <a href="https://eudml.org/doc/267245">alternative link</a>.
%H A371757 Wikipedia, <a href="http://en.wikipedia.org/wiki/Pseudoprime">Pseudoprime</a>.
%H A371757 <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>.
%t A371757 f[p_, k_] := (4^(p^(k + 1)) + 1)/(4^(p^k) + 1);
%t A371757 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]
%Y A371757 Subsequence of A001567.
%Y A371757 Cf. A210454, A293626.
%K A371757 nonn
%O A371757 1,1
%A A371757 _Amiram Eldar_, Apr 05 2024