A260080 Least positive integer k such that prime(k*n)^2 - 2 = prime(i*n)*prime(j*n) for some integers 0 < i < j.

5, 18, 18, 9, 115, 208, 69, 373, 68, 430, 8, 214, 57, 1887, 1255, 295, 880, 542, 5612, 767, 1562, 40, 853, 884, 753, 4332, 4750, 6077, 799, 1394, 639, 5442, 4785, 440, 7417, 1290, 15830, 27745, 3927, 5701, 1891, 22008, 8243, 6031, 9172, 5949, 43286, 20778, 9876, 12472

Offset

1,1

Comments

Conjecture: a(n) exists for any n > 0.

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..105

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

Example

a(1) = 5 since prime(5*1)^2-2 = 11^2-2 = 119 = 7*17 = prime(4*1)*prime(7*1).
a(66) = 149073 since prime(149073*66)^2-2 = 176365951^2-2 = 31104948672134399 = 3160879*9840600881 = prime(3448*66)*prime(9840600881*66).

Mathematica

Dv[n_]:=Divisors[Prime[n]^2-2]
L[n_]:=Length[Dv[n]]
P[k_, n_]:=L[k*n]==4&&PrimeQ[Part[Dv[k*n], 2]]&&Mod[PrimePi[Part[Dv[k*n], 2]], n]==0&&PrimeQ[Part[Dv[k*n], 3]]&&Mod[PrimePi[Part[Dv[k*n], 3]], n]==0
Do[k=0; Label[bb]; k=k+1; If[P[k, n], Goto[aa]]; Goto[bb]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 50}]

Crossrefs

Cf. A000040, A062326, A253257, A257926, A257938, A260078.

Sequence in context: A120087 A353690 A180135 * A178365 A345912 A022142

Adjacent sequences: A260077 A260078 A260079 * A260081 A260082 A260083

Keyword

nonn

Author

Zhi-Wei Sun, Jul 15 2015

Status

approved