A260078 Least positive integer k such that prime(k*n)-1+(prime(h*n)-1) = prime(i*n)-1 and prime(k*n)-1-(prime(h*n)-1) = prime(j*n)-1 for some positive integers h,i,j.

3, 3, 15, 5, 25, 29, 32, 20, 41, 87, 17, 61, 18, 100, 58, 10, 82, 82, 45, 74, 166, 20, 28, 338, 18, 35, 159, 290, 64, 29, 353, 311, 75, 41, 42, 492, 107, 155, 77, 364, 100, 330, 145, 474, 502, 332, 227, 553, 238, 92, 121, 597, 338, 339, 452, 164, 239, 832, 221, 243

Offset: 1

Author

Zhi-Wei Sun, Jul 15 2015

Keywords

• nonn

Comments

Conjecture: a(n) exists for any n > 0. In general, if m and n > 0 are integers with gcd(6,m) = 1, then the set {prime(k*n)+m: k = 1,2,3,...} contains two distinct elements x and y with x+y and x-y also in the set.

Examples

			a(2) = 3 since prime(3*2)-1+(prime(2*2)-1) = 12+6 = 18 = prime(4*2)-1, and prime(3*2)-1-(prime(2*2)-1) = 12-6 = 6 = prime(2*2)-1.
a(3) = 15 since prime(15*3)-1+(prime(12*3)-1) = 196+150 = 346 = prime(23*3)-1, and prime(15*3)-1-(prime(12*3)-1) = 196 -150 = 46 = prime(5*3)-1.
a(200) = 3319 since prime(3319*200)-1+(prime(2821*200)-1) = 9987120+8389110 = 18376230 = prime(5869*200)-1, and prime(3319*200)-1-(prime(2821*200)-1) = 9987120-8389110 = 1598010 = prime(605*200)-1.
		

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..200
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000040, A257926, A257938.

Programs

Mathematica

f[n_]:=Prime[n]-1
PQ[n_,p_]:=PrimeQ[p]&&Mod[PrimePi[p],n]==0
Do[k=0;Label[bb];k=k+1;Do[If[PQ[n,f[k*n]+f[j*n]+1]&&PQ[n,f[k*n]-f[j*n]+1],Goto[aa]],{j,1,k-1}];Goto[bb];
Label[aa];Print[n," ",k];Continue,{n,1,60}]