A196779 a(n) is the smallest number m such that no prime takes the form of n^m+/-n^k+/-1, while 0 <= k < m and m > 1.

1147, 113, 113, 400, 866, 131, 399, 32, 26, 29, 23, 58, 77, 21, 42, 3, 817, 4, 2, 37, 80, 29, 181, 39, 120, 382, 76, 5, 29, 20, 48, 19, 36, 7, 43, 7, 62, 22, 7, 43, 5, 17, 23, 44, 52, 137, 103, 2, 5, 49, 31, 10, 30, 5, 25, 25, 49, 10, 72, 50, 13, 4, 7, 6

Offset: 5

Author

Lei Zhou, Oct 06 2011

Keywords

• nonn

Comments

Conjecture: a(n) has finite value when a>4

already tested: a(4)>2364; a(3)>7399; and a(2)>9594.

Hypothesis is that a(2), a(3), and a(4) are infinite.

Mathematica program ran about an hour and gave the first 96 items.

When n is larger, a(n) tends to be 2 for most of n.

Examples

			n=5, there is no prime number in the form of 5^1147+/-5^k+/-1 for 0 <= k < 1147
		

Crossrefs

Cf. A196697, A196698, A196778.

Programs

Mathematica

Table[i = 1;  While[i++; c1 = b^i; cs = {};
  Do[c2 = b^j; cp = c1 + c2 + 1;
   If[PrimeQ[cp], cs = Union[cs, {cp}]];
   cp = c1 + c2 - 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
   cp = c1 - c2 + 1; If[PrimeQ[cp], cs = Union[cs, {cp}]];
   cp = c1 - c2 - 1;
   If[PrimeQ[cp], cs = Union[cs, {cp}]], {j, 0, i - 1}];
  ct = Length[cs]; ct > 0]; i, {b, 5, 100}]