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A265653 Integers k such that (k-1)^3 + 1 is a Fermat pseudoprime to base 2 (A001567).

%I A265653 #48 Apr 25 2024 21:09:43
%S A265653 13,37,139,271,547,4801,7561,12841,14701,358201,678481,16139971,
%T A265653 22934101,55058581,59553721,74371321,113068381,116605861,242699311,
%U A265653 997521211,1592680321,1652749201,3190927741,5088964801,6974736757,9214178821
%N A265653 Integers k such that (k-1)^3 + 1 is a Fermat pseudoprime to base 2 (A001567).
%C A265653 Corresponding Fermat pseudoprimes to base 2 are 1729, 46657, 2628073, 19683001, 162771337, 110592000001, 432081216001, ...
%C A265653 There is only one composite term up to 10^10: 14701. It also appears in A265628 (see comments). Can we say that if there is a Fermat pseudoprime to base 2 of the form (k-1)^3 + 1, k is a prime number most of the time? Are there other composite terms like 14701?
%F A265653 a(n) = A270840(n) + 1.
%e A265653 13 is a term because (13-1)^3 + 1 = 1729, which is a Fermat pseudoprime to base 2.
%e A265653 37 is a term because (37-1)^3 + 1 = 46657, which is a Fermat pseudoprime to base 2.
%t A265653 Select[Range[10^6], ! PrimeQ@ # && PowerMod[2, (# - 1), #] == 1 &@((# - 1)^3 + 1) &] (* _Michael De Vlieger_, Dec 12 2015, after _Farideh Firoozbakht_ at A001567 *)
%o A265653 (PARI) is(n) = {Mod(2, n)^n==2 & !isprime(n) & n>1};
%o A265653 for(n=1, 1e10, if(is((n-1)^3+1), print1(n, ", ")));
%Y A265653 Cf. A000040, A001567, A265628, A270840.
%K A265653 nonn
%O A265653 1,1
%A A265653 _Altug Alkan_, Dec 12 2015