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

1,1

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.

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.

Example

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.

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}]

Crossrefs

Cf. A000040, A257926, A257938.

Sequence in context: A100347 A165405 A179857 * A163590 A344850 A114320

Adjacent sequences: A260075 A260076 A260077 * A260079 A260080 A260081

Keyword

nonn

Author

Zhi-Wei Sun, Jul 15 2015

Status

approved