%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^22 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^21 (or 4*p^2+1, or p^2+p+1) with p prime.
%D ZhiWei 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 ChinaJapan Seminar (Fukuoka, Oct. 28  Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169187.
%H ZhiWei Sun, <a href="/A253257/b253257.txt">Table of n, a(n) for n = 1..800</a>
%H ZhiWei 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 = 2^22 with 2 prime.
%e a(6) = 12 since prime(12*6) = 359 = 19^22 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($k*$n1)}; 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 _ZhiWei Sun_, Jul 05 2015
