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A286680 Smallest nonnegative m such that (1 + n)^(2^m) + n is not prime. 2
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 (list; graph; refs; listen; history; text; internal format)
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 A109430 A085917

Adjacent sequences:  A286677 A286678 A286679 * A286681 A286682 A286683

KEYWORD

nonn

AUTHOR

Juri-Stepan Gerasimov, May 12 2017

STATUS

approved

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Last modified October 15 03:16 EDT 2019. Contains 328025 sequences. (Running on oeis4.)