A229791 Integers generated by (a^k+b^k)/(a+b) for all possible positive integer values of a,b,k with b>a.

1, 3, 5, 7, 10, 11, 12, 13, 15, 17, 19, 20, 21, 25, 26, 27, 28, 29, 30, 31, 34, 35, 37, 39, 40, 41, 43, 45, 48, 49, 50, 51, 52, 53, 55, 57, 58, 60, 61, 63, 65, 67, 68, 70, 73, 74, 75, 76, 78, 79, 80, 82, 84, 85, 87, 89, 90, 91, 93, 95, 97, 100, 101, 102, 103

Offset: 1

Author

Robert Price, Sep 29 2013

Keywords

• nonn

Comments

This form, (a^k+b^k)/(a+b), is a generalization of the Fermat numbers.

Not all integers are in this set.

See A227979 for the complement of this sequence.

Links

• Robert Price, Table of n, a(n) for n = 1..134
• J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
• H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
• H. Lifchitz, Mersenne and Fermat primes field
• Eric Weisstein's World of Mathematics, Repunit.

Crossrefs

A few of the sequences using this form that identify primes are A000978, A007658, A057469, A128066, A057171, A082387, A122853, A128335.

Programs

Mathematica

limit=105; lst = {}; Do[p = (a^k + b^k)/(a + b); If[p <= limit && IntegerQ[p], AppendTo[lst, p]], {k, Log[2,3*limit+1]}, {b, 2, limit*2}, {a, b-1}]; Union[lst]