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A257938
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Least positive integer k such that prime(k*n) - 1 = (prime(i*n)-1)*(prime(j*n)-1) for some integers 0 < i < j < k.
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8
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6, 3, 8, 71, 12, 14, 105, 221, 24, 499, 261, 612, 1341, 175, 917, 549, 1351, 2303, 2273, 4767, 364, 1395, 1390, 1431, 6481, 2479, 918, 2412, 17783, 3178, 2994, 7538, 3409, 1361, 9645, 3454, 9197, 7074, 10418, 6059, 36235, 182, 1910, 4648, 1130, 695, 3973, 10839, 8647, 7942
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OFFSET
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1,1
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COMMENTS
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Conjecture: a(n) exists for any n > 0. In general, for any nonzero integer m and positive integer n, the set {prime(k*n)+m: k = 1,2,3,...} always contains three distinct elements x, y and z with x*y = z.
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REFERENCES
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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.
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LINKS
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EXAMPLE
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a(1) = 6 since prime(6*1)-1 = 12 = 2*6 = (prime (2*1)-1)*(prime(4*1)-1).
a(4) = 71 since prime(71*4)-1 = 1860 = 6*310 = (prime(1*4)-1)*(prime(16*4)-1).
a(41) = 36235 since prime(36235*41)-1 = 23634312 = 676*34962 = (prime(3*41)-1)*(prime(91*41)-1).
a(69) = 64999 since prime(64999*69)-1 = 76643820 = 4590*16698 = (prime(9*69)-1)*(prime(28*69)-1).
a(77) = 137789 since prime(137789*77)-1 = 191037600 = 2028*94200 = (prime(4*77)-1)*(prime(118*77)-1).
a(99) = 167708 since prime(167708*99)-1 = 306849088 = 10528*29146 = (prime(13*99)-1)*(prime(32*99)-1).
a(189) = 951492 since prime(951492*189)-1 = 3776304996 = 4126*915246 = (prime(3*189)-1)*(prime(383*189)-1).
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MATHEMATICA
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Dv[n_]:=Divisors[Prime[n]-1]
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, 2, L[k*n]/2}]; Goto[bb]; Label[aa]; Print[n, " ", k]; Continue, {n, 1, 50}]
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PROG
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(PARI) a(n)={my(i, j, k=3); while(1, for(j=2, k-1, for(i=1, j-1, if(prime(k*n) - 1 == (prime(i*n)-1)*(prime(j*n)-1), break(3)); )); k++); return(k); }
main(size)={return(vector(size, n, a(n))); } /* Anders Hellström, Jul 13 2015 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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