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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
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 (list; graph; refs; listen; history; text; internal format)
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^2-2 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^2-1 (or 4*p^2+1, or p^2+p+1) with p prime.

REFERENCES

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.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..800

Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

EXAMPLE

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

a(6) = 12 since prime(12*6) = 359 = 19^2-2 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($k*$n-1)}; say "$n $k"; } # Dana Jacobsen, Dec 15 2015

CROSSREFS

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

Sequence in context: A126744 A122078 A126736 * A127412 A304791 A306646

Adjacent sequences:  A253254 A253255 A253256 * A253258 A253259 A253260

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jul 05 2015

STATUS

approved

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Last modified August 19 05:11 EDT 2019. Contains 326109 sequences. (Running on oeis4.)