A283619 a(n) = (conjectured) smallest positive integer k which is neither of the form p + n^x nor of the form p - n^x with x >= 0 and p prime, where gcd(k, n) = 1 and gcd(k^2-1, n-1) = 1.

30666137, 3902132276156, 2473929, 1015214, 464437, 40743218950116, 47, 2344, 61863, 32660, 4367, 7974, 11, 2021170066180678, 92343, 784, 571, 2364594, 13, 20450, 136113, 2596, 176011, 262638, 3223, 512, 59217, 26, 18973, 6360528, 23, 11848, 99, 292226, 832573

Offset: 2

Author

Arkadiusz Wesolowski, Mar 12 2017

Keywords

• nonn

Comments

The definition is similar to that for A123159, but considering "p + n^x" and "p - n^x".

What does "conjectured" mean? A positive integer k is a candidate if:

1) gcd(k, n) = 1,

2) gcd(k^2-1, n-1) = 1,

3) every term in the sequence k + n^x is divisible by one of the prime numbers of a covering set,

4) all numbers of the form k - n^x are composite, k > n^x + 1, x >= 0.

The main problem is to prove that the given terms are indeed correct.

A quick search showed that a(8) = 47, a(14) = 11, a(20) = 13, a(27) = 512, a(29) = 26, a(32) = 23, a(34) = 99.

This is an interesting sequence: it leads to new classes of numbers. For example, the integer 30666137 is probably the smallest number that is simultaneously a Polignac number and a Sierpinski number.

Crossrefs

Cf. A076336, A123159, A263644, A283622.