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

13, 37, 139, 271, 547, 4801, 7561, 12841, 14701, 358201, 678481, 16139971, 22934101, 55058581, 59553721, 74371321, 113068381, 116605861, 242699311, 997521211, 1592680321, 1652749201, 3190927741, 5088964801, 6974736757, 9214178821

Offset: 1

Author

Altug Alkan, Dec 12 2015

Keywords

• nonn

Comments

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

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?

Examples

			13 is a term because (13-1)^3 + 1 = 1729, which is a Fermat pseudoprime to base 2.
37 is a term because (37-1)^3 + 1 = 46657, which is a Fermat pseudoprime to base 2.
		

Crossrefs

Cf. A000040, A001567, A265628, A270840.

Programs

Mathematica

Select[Range[10^6], ! PrimeQ@ # && PowerMod[2, (# - 1), #] == 1 &@((# - 1)^3 + 1) &] (* Michael De Vlieger, Dec 12 2015, after Farideh Firoozbakht at A001567 *)

PARI

is(n) = {Mod(2, n)^n==2 & !isprime(n) & n>1};
for(n=1, 1e10, if(is((n-1)^3+1), print1(n, ", ")));

Formula

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