A265653 - OEIS

%I #48 Apr 25 2024 21:09:43

%S 13,37,139,271,547,4801,7561,12841,14701,358201,678481,16139971,

%T 22934101,55058581,59553721,74371321,113068381,116605861,242699311,

%U 997521211,1592680321,1652749201,3190927741,5088964801,6974736757,9214178821

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

%C Corresponding Fermat pseudoprimes to base 2 are 1729, 46657, 2628073, 19683001, 162771337, 110592000001, 432081216001, ...

%C 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 a(n) = A270840(n) + 1.

%e 13 is a term because (13-1)^3 + 1 = 1729, which is a Fermat pseudoprime to base 2.

%e 37 is a term because (37-1)^3 + 1 = 46657, which is a Fermat pseudoprime to base 2.

%t Select[Range[10^6], ! PrimeQ@ # && PowerMod[2, (# - 1), #] == 1 &@((# - 1)^3 + 1) &] (* _Michael De Vlieger_, Dec 12 2015, after _Farideh Firoozbakht_ at A001567 *)

%o (PARI) is(n) = {Mod(2, n)^n==2 & !isprime(n) & n>1};

%o for(n=1, 1e10, if(is((n-1)^3+1), print1(n, ", ")));

%Y Cf. A000040, A001567, A265628, A270840.

%K nonn

%O 1,1

%A _Altug Alkan_, Dec 12 2015