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%I #42 May 26 2017 22:55:08
%S 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,
%T 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,
%U 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
%N Smallest nonnegative m such that (1 + n)^(2^m) + n is not prime.
%C Nonprimes: 1, 4294967297, 43046723, 259, 9, 1679621, 55, 15, 43046729, 100000009, 21, 155, 25, 27, 50639, 18446744073709551631, 33, 35, ...
%C Conjecture: a(n) <= 6 for all n.
%C 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
%C 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
%C From _Robert G. Wilson v_, May 18 2017: (Start)
%C m=
%C 0: 0, 4, 7, 10, 12, 13, 16, 17, 19, 22, 24, 25, 27, 28, 31, 32, 34, 37, 38, etc.;
%C 1: 6, 11, 21, 26, 33, 35, 36, 39, 41, 48, 50, 51, 56, 68, 74, 78, 81, 83, etc.;
%C 2: 3, 14, 18, 20, 23, 29, 44, 54, 63, 65, 69, 75, 95, 99, 113, 114, 125, etc.;
%C 3: 5, 8, 9, 30, 53, 119, 230, 308, 329, 350, 624, 638, 779, 785, 813, 1110, etc.;
%C 4: 2, 15, 2100, 4223, 4773, 7868, 8744, 9339, 9540, 13178, 14589, 15884, etc.;
%C 5: 1, 1432578, 1627035, 1737054, 1888094, 1959638, 2176139, 3172304, 3425069, etc.;
%C 6: 54131988, 177386619, 229940778, 846372674, 2124404844, 2367307088, 2539775055, etc.;
%C (End)
%H Robert G. Wilson v, <a href="/A286680/b286680.txt">Table of n, a(n) for n = 0..10000</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Bunyakovsky_conjecture">Bunyakovsky conjecture</a>.
%e a(0) = 0 because (1 + 0)^(2^0) + 0 = 1 is not prime.
%p f:= proc(n) local k;
%p for k from 0 while isprime((1+n)^(2^k)+n) do od:
%p k;
%p end proc:
%p map(f, [$0..100]); # _Robert Israel_, May 17 2017
%t 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 *)
%o (PARI) a(n) = {my(m = 0); while (isprime((1 + n)^(2^m) + n), m++); m;} \\ _Michel Marcus_, May 19 2017
%Y Cf. A019434, A047845, A057726, A160027.
%K nonn
%O 0,2
%A _Juri-Stepan Gerasimov_, May 12 2017
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