A259628 Numbers m with m-1, m+1, prime(m)+2, prime(m)-m, prime(m)+m, m*prime(m)-1 and m*prime(m)+1 all prime.

2523708, 6740478, 6759030, 14655522, 22885698, 28384200, 44630148, 71742300, 87002328, 87466500, 89842200, 153110622, 153647490, 184373490, 283232040, 312124920, 366318960, 408770670, 412216920, 439429128, 456486030, 486730398, 517602600, 606159978, 607942848, 675661080, 855983352, 869593998, 923864562, 971400672

Offset: 1

Author

Zhi-Wei Sun, Jul 01 2015

Keywords

• nonn

Comments

Conjecture: The sequence contains infinitely many terms.

This is stronger than the conjectures in A232861 and A259539.

Examples

			a(1) = 2523708 since the seven numbers 2523707, 2523709, prime(2523708)+2 = 41578739+2 = 41578741, prime(2523708)-2523708 = 41578739-2523708 = 39055031, prime(2523708)+2523708 = 41578739+2523708 = 44102447, 2523708*prime(2523708)-1 = 2523708*41578739-1 = 104932596244211 and 2523708*prime(2523708)+1 = 2523708*41578739+1 = 104932596244213 are all 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..160
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A014574, A232861, A259539.

Programs

Mathematica

TW[n_]:=PrimeQ[n-1]&&PrimeQ[n+1]
n=0;Do[If[PrimeQ[Prime[k]+2]&&PrimeQ[Prime[Prime[k]+1]+2]&&PrimeQ[Prime[Prime[k]+1]-Prime[k]-1]&&PrimeQ[Prime[Prime[k]+1]+Prime[k]+1]&&TW[(Prime[k]+1)Prime[Prime[k]+1]],n=n+1;Print[n," ",Prime[k]+1]],{k,1,5*10^7}]
allprQ[n_]:=Module[{p=Prime[n]},AllTrue[{n-1,n+1,p+2,p-n,p+n,n*p-1, n*p+1}, PrimeQ]]; Select[Range[98*10^7],allprQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 26 2016 *)