A260080 -id:A260080 - cp's OEIS frontend

A260082 Least positive integer k such that (prime(kn)-1)^2 = (prime(in)-1)(prime(jn)-1) for some integers 0 < i < j.

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

Offset: 1

Zhi-Wei Sun, Jul 15 2015

nonn

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.

			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).

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.

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.

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

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}]

A260121 Least positive integer k such that prime(kn)^2 - 2 = prime(jn) for some j > 0.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 45, 1, 15, 34, 9, 146, 63, 128, 9, 20, 79, 45, 242, 50, 44, 71, 103, 181, 98, 208, 5, 180, 162, 299, 710, 10, 3, 388, 144, 427, 225, 121, 79, 25, 580, 230, 471, 46, 3, 1040, 11, 224, 305, 56, 1163, 104, 93, 193, 55, 90, 88, 521, 898, 218

Offset: 1

Zhi-Wei Sun, Jul 17 2015

nonn

The conjecture in A260120 implies that a(n) exists for any n > 0, which is stronger than the conjecture in A253257.

			a(5) = 4 since prime(4*5)^2-2 = 71^2-2 = 5039 = prime(135*5).

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.

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.

Cf. A000040, A062326, A253257, A260080, A260120.

P[n_,p_]:=PrimeQ[p]&&Mod[PrimePi[p],n]==0
Do[k=0;Label[bb];k=k+1; If[P[n,Prime[k*n]^2-2],Goto[aa]];Goto[bb];Label[aa];Print[n, " ", k];Continue,{n,1,60}]

A258580 Least positive integer k such that (prime(jn)+prime(kn))/2 = prime(i*n)^2 for some integers i > 0 and 0 < j < k.

Original entry on oeis.org

3, 9, 4, 127, 98, 133, 55, 78, 65, 85, 375, 109, 251, 283, 105, 462, 681, 149, 156, 213, 525, 209, 205, 381, 757, 313, 252, 615, 61, 737, 478, 1754, 406, 1197, 131, 420, 492, 503, 127, 119, 549, 1748, 95, 442, 2740, 555, 677, 1258, 163, 816, 1649, 710, 203, 126, 628, 582, 1004, 135, 837, 1000

Offset: 1

Zhi-Wei Sun, Jul 15 2015

nonn

Conjecture: a(n) exists for any n > 0. In general, for any positive integers a, m and n, there are integers i,j,k > 0 with i > j such that (prime(i*n)+prime(j*n))/2 (or (prime(i*n)-prime(j*n))/2) is equal to a*prime(k*n)^m.

			a(1) = 3 since (prime(2*1)+prime(3*1))/2 = (3+5)/2 = 2^2 = prime(1*1)^2.
a(158) = 8405 since (prime(778*158)+prime(8405*158))/2 = (1625551+20967091)/2 = 3361^2 = prime(3*158)^2.

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.

Zhi-Wei Sun, Table of n, a(n) for n = 1..160
Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Cf. A000040, A257926, A257938, A260078, A260080, A260082.

PQ[n_,m_]:=PrimeQ[Sqrt[m]]&&Mod[PrimePi[Sqrt[m]],n]==0
Do[k=0;Label[bb];k=k+1;Do[If[PQ[n,(Prime[k*n]+Prime[j*n])/2],Goto[aa]];Continue,{j,1,k-1}];Goto[bb];
Label[aa];Print[n," ",k];Continue,{n,1,60}]

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A260082 Least positive integer k such that (prime(kn)-1)^2 = (prime(in)-1)(prime(jn)-1) for some integers 0 < i < j.

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A260121 Least positive integer k such that prime(kn)^2 - 2 = prime(jn) for some j > 0.

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A258580 Least positive integer k such that (prime(jn)+prime(kn))/2 = prime(i*n)^2 for some integers i > 0 and 0 < j < k.

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cp's OEIS Frontend

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.

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A260121 Least positive integer k such that prime(k*n)^2 - 2 = prime(j*n) for some j > 0.

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A258580 Least positive integer k such that (prime(j*n)+prime(k*n))/2 = prime(i*n)^2 for some integers i > 0 and 0 < j < k.

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A260082 Least positive integer k such that (prime(kn)-1)^2 = (prime(in)-1)(prime(jn)-1) for some integers 0 < i < j.

A260121 Least positive integer k such that prime(kn)^2 - 2 = prime(jn) for some j > 0.

A258580 Least positive integer k such that (prime(jn)+prime(kn))/2 = prime(i*n)^2 for some integers i > 0 and 0 < j < k.