A283619 - OEIS

%I #11 Apr 01 2017 17:59:23

%S 30666137,3902132276156,2473929,1015214,464437,40743218950116,47,2344,

%T 61863,32660,4367,7974,11,2021170066180678,92343,784,571,2364594,13,

%U 20450,136113,2596,176011,262638,3223,512,59217,26,18973,6360528,23,11848,99,292226,832573

%N 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.

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

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

%C 1) gcd(k, n) = 1,

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

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

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

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

%C 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.

%C 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.

%Y Cf. A076336, A123159, A263644, A283622.

%K nonn

%O 2,1

%A _Arkadiusz Wesolowski_, Mar 12 2017