A237367 -id:A237367 - cp's OEIS frontend

A237413 Number of ways to write n = k + m with k > 0 and m > 0 such that p(k)^2 - 2, p(m)^2 - 2 and p(p(m))^2 - 2 are all prime, where p(j) denotes the j-th prime.

0, 1, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 3, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 5, 3, 1, 3, 3, 3, 3, 3, 1, 3, 1, 2, 2, 5, 2, 3, 3, 5, 2, 5, 7, 3, 3, 4, 5, 5, 5, 4, 4, 5, 2, 3, 4, 7, 5, 3, 4, 8, 6, 5, 4, 6, 5, 4, 2, 6, 5, 6, 5, 2, 6, 7

Offset: 1

Zhi-Wei Sun, Feb 07 2014

nonn

Conjecture: a(n) > 0 for all n > 1.
This conjecture was motivated by the "Super Twin Prime Conjecture".
See A237414 for primes q with q^2 - 2 and p(q)^2 - 2 both prime.

			a(7) = 1 since 7 = 6 + 1 with p(6)^2 - 2 = 13^2 - 2 = 167, p(1)^2 - 2 = 2^2 - 2 = 2 and p(p(1))^2 - 2 = p(2)^2 - 2 = 3^2 - 2 = 7 are all prime.
a(516) = 1 since 516 = 473 + 43 with p(473)^2 - 2 = 3359^2 - 2 = 11282879, p(43)^2 - 2 = 191^2 - 2 = 36479 and p(p(43))^2 - 2 = p(191)^2 - 2 = 1153^2 - 2 = 1329407 all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Super Twin Prime Conjecture, a message to Number Theory List, Feb. 6, 2014.
Z.-W. Sun, Problems on combinatorial properties of primes, arXiv:1402.6641, 2014

Cf. A000040, A049002, A062326, A218829, A237348, A237367, A237414.

pq[k_]:=PrimeQ[Prime[k]^2-2]
a[n_]:=Sum[If[pq[k]&&pq[n-k]&&pq[Prime[n-k]],1,0],{k,1,n-1}]
Table[a[n],{n,1,80}]

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.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Jul 05 2015

nonn

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.

			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.

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.

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.

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

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}]

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

cp's OEIS Frontend

A237413 Number of ways to write n = k + m with k > 0 and m > 0 such that p(k)^2 - 2, p(m)^2 - 2 and p(p(m))^2 - 2 are all prime, where p(j) denotes the j-th prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica