

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



1, 2, 3, 1, 3, 12, 47, 9, 1, 100, 502, 6, 3, 1817, 1, 362, 3141, 4, 104, 50, 14157, 251, 222, 3, 27, 76, 25, 5423, 416, 73, 28764, 181, 488, 3860, 1249, 2, 138, 52, 1, 25, 8734, 65719, 7089, 214, 15, 111, 7, 990, 6254, 20, 1047, 38, 367, 880, 435, 3712, 3287, 208, 5194, 598
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OFFSET

1,2


COMMENTS

Conjecture: a(n) > 0 for all n > 0.
This is stronger than the conjecture that there are infinitely many primes of the form p^22 with p prime.
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.


REFERENCES

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.


LINKS



EXAMPLE

a(1) = 1 since prime(1*1) = 2 = 2^22 with 2 prime.
a(6) = 12 since prime(12*6) = 359 = 19^22 with 19 prime.


MATHEMATICA

SQ[n_]:=IntegerQ[Sqrt[n]]&&PrimeQ[Sqrt[n]]
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}]


PROG

(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


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



