cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

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

Original entry on oeis.org

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

Views

Author

Zhi-Wei Sun, Jul 15 2015

Keywords

Comments

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

Examples

			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).

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

Crossrefs

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

Programs

  • 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}]

A260121 Least positive integer k such that prime(k*n)^2 - 2 = prime(j*n) 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

Views

Author

Zhi-Wei Sun, Jul 17 2015

Keywords

Comments

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

Examples

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

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

Crossrefs

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

Programs

  • Mathematica
    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}]
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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