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A253257 Least positive integer k such that prime(k*n) has the form p^2 - 2 with p prime, or 0 if no such k exists. 3

%I

%S 1,2,3,1,3,12,47,9,1,100,502,6,3,1817,1,362,3141,4,104,50,14157,251,

%T 222,3,27,76,25,5423,416,73,28764,181,488,3860,1249,2,138,52,1,25,

%U 8734,65719,7089,214,15,111,7,990,6254,20,1047,38,367,880,435,3712,3287,208,5194,598

%N Least positive integer k such that prime(k*n) has the form p^2 - 2 with p prime, or 0 if no such k exists.

%C Conjecture: a(n) > 0 for all n > 0.

%C This is stronger than the conjecture that there are infinitely many primes of the form p^2-2 with p prime.

%C I also conjecture that for any positive integer n there is a positive integer k such that prime(k*n) has the form 2*p^2-1 (or 4*p^2+1, or p^2+p+1) with p prime.

%D Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.

%H Zhi-Wei Sun, <a href="/A253257/b253257.txt">Table of n, a(n) for n = 1..800</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.

%e a(1) = 1 since prime(1*1) = 2 = 2^2-2 with 2 prime.

%e a(6) = 12 since prime(12*6) = 359 = 19^2-2 with 19 prime.

%t SQ[n_]:=IntegerQ[Sqrt[n]]&&PrimeQ[Sqrt[n]]

%t Do[k=0;Label[bb];k=k+1;If[SQ[Prime[k*n]+2],Goto[aa],Goto[bb]];Label[aa];Print[n, " ", k];Continue,{n,1,60}]

%o (Perl) use ntheory ":all"; use Math::Prime::Util::PrimeArray qw/$probj/; my %v; forprimes { undef $v{$_*$_-2} } 4e7; for my $n (1..800) { my $k=1; $k++ until exists $v{$probj->FETCH($k*$n-1)}; say "$n $k"; } # _Dana Jacobsen_, Dec 15 2015

%Y Cf. A000040, A049002, A060429, A060800, A062326, A179262, A237367, A237413, A259731.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Jul 05 2015

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