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

0,2

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)

Links

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

Wikipedia, Bunyakovsky conjecture.

Example

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

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 *)

Prog

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

Crossrefs

Cf. A019434, A047845, A057726, A160027.

Sequence in context: A102220 A279151 A224228 * A201680 A349409 A109430

Adjacent sequences: A286677 A286678 A286679 * A286681 A286682 A286683

Keyword

nonn

Author

Juri-Stepan Gerasimov, May 12 2017

Status

approved