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A260082 Least positive integer k such that (prime(k*n)-1)^2 = (prime(i*n)-1)*(prime(j*n)-1) for some integers 0 < i < j. 4
2, 2, 2, 21, 9, 10, 12, 14, 47, 32, 32, 171, 177, 175, 64, 187, 330, 206, 77, 467, 4, 126, 127, 355, 279, 982, 249, 1930, 105, 109, 659, 801, 269, 777, 703, 125, 819, 1347, 904, 1153, 549, 2344, 757, 1301, 1793, 303, 105, 3168, 2645, 3055, 110, 1619, 1580, 2423, 220, 965, 1397, 84, 988, 322 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: a(n) exists for any n > 0. In general, for any nonzero integer m and positive integer n there are distinct positive integers i,j,k such that (prime(i*n)+m)*(prime(j*n)+m) = (prime(k*n)+m)^2.

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..100

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

EXAMPLE

a(4) = 21 since (prime(21*4)-1)^2 = 432^2 = 18*10368 = (prime(2*4)-1)*(prime(318*4)-1).

a(61) = 15160 since (prime(15160*61)-1)^2 = 14242116^2 = 47316*4286876916 = (prime(80*61)-1)*(prime(3326491*61)-1).

MATHEMATICA

Dv[n_]:=Divisors[(Prime[n]-1)^2]

L[n_]:=Length[Dv[n]]

P[k_, n_, i_]:=PrimeQ[Part[Dv[k*n], i]+1]&&Mod[PrimePi[Part[Dv[k*n], i]+1], n]==0

Do[k=0; Label[bb]; k=k+1; Do[If[P[k, n, i]&&P[k, n, L[k*n]-i+1], Goto[aa]], {i, 1, L[k*n]/2}]; Goto[bb]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 60}]

CROSSREFS

Cf. A000040, A257926, A257928, A257938, A260078, A260080.

Sequence in context: A099640 A140283 A067097 * A098915 A095855 A157979

Adjacent sequences:  A260079 A260080 A260081 * A260083 A260084 A260085

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Jul 15 2015

STATUS

approved

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Last modified August 26 02:19 EDT 2019. Contains 326324 sequences. (Running on oeis4.)