A286680 Smallest nonnegative m such that (1 + n)^(2^m) + n is not prime.

0, 5, 4, 2, 0, 3, 1, 0, 3, 3, 0, 1, 0, 0, 2, 4, 0, 0, 2, 0, 2, 1, 0, 2, 0, 0, 1, 0, 0, 2, 3, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 1, 1, 0, 3, 2, 0, 1, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0, 0, 0, 1, 2, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 2

Offset: 0

Author

Juri-Stepan Gerasimov, May 12 2017

Keywords

• nonn

Comments

Nonprimes: 1, 4294967297, 43046723, 259, 9, 1679621, 55, 15, 43046729, 100000009, 21, 155, 25, 27, 50639, 18446744073709551631, 33, 35, ...

Conjecture: a(n) <= 6 for all n.

This conjecture would contradict the generalized Bunyakovsky conjecture. That is, the polynomials (1+n)^k+n for k=0..6 satisfy the conditions for that conjecture, and so there should be some n for which all seven are prime. - Robert Israel, May 17 2017

Smallest k such that (1 + k)^(2^n) + k is not prime: 0, 6, 3, 5, 2, 1, 54131988 (conjecturally finite). Last term found by Robert G. Wilson v, May 14 2017

From Robert G. Wilson v, May 18 2017: (Start)

m=

0: 0, 4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, 32, 34, 37, 38, etc.;

1: 6, 11, 21, 26, 33, 35, 36, 39, 41, 48, 50, 51, 56, 68, 74, 78, 81, 83, etc.;

2: 3, 14, 18, 20, 23, 29, 44, 54, 63, 65, 69, 75, 95, 99, 113, 114, 125, etc.;

3: 5, 8, 9, 30, 53, 119, 230, 308, 329, 350, 624, 638, 779, 785, 813, 1110, etc.;

4: 2, 15, 2100, 4223, 4773, 7868, 8744, 9339, 9540, 13178, 14589, 15884, etc.;

5: 1, 1432578, 1627035, 1737054, 1888094, 1959638, 2176139, 3172304, 3425069, etc.;

6: 54131988, 177386619, 229940778, 846372674, 2124404844, 2367307088, 2539775055, etc.;

(End)

Examples

			a(0) = 0 because (1 + 0)^(2^0) + 0 = 1 is not prime.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 0..10000
• Wikipedia, Bunyakovsky conjecture.

Crossrefs

Cf. A019434, A047845, A057726, A160027.

Programs

Maple

f:= proc(n) local k;
  for k from 0 while isprime((1+n)^(2^k)+n) do od:
  k;
end proc:
map(f, [$0..100]); # Robert Israel, May 17 2017

Mathematica

f[n_] := Block[{k = 0}, While[ PrimeQ[(1 + n)^(2^k) + n], k++]; k]; Array[f, 105, 0] (* Robert G. Wilson v, May 14 2017 *)

PARI

a(n) = {my(m = 0); while (isprime((1 + n)^(2^m) + n), m++); m;} \\ Michel Marcus, May 19 2017