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.

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A261583 Least positive integer k such that prime(prime(prime(k)))+ prime(prime(prime(k*n))) = 2*prime(prime(p)) for some prime p.

Original entry on oeis.org

1, 755, 4648, 1335, 1096, 14708, 5964, 636, 1063, 13019, 9808, 2776, 2580, 2797, 6421, 1573, 2432, 4790, 862, 1855, 566, 2145, 18554, 35634, 5264, 1293, 39402, 1445, 2397, 17930, 586, 2526, 24571, 18403, 5480, 366, 5159, 9710, 179, 4469, 6757, 7866, 263, 1701, 2941, 477, 5032, 10705, 3494, 8597, 953, 11954, 2586, 689, 9456, 1309, 8651, 12538, 4106, 13762
Offset: 1

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Author

Zhi-Wei Sun, Aug 25 2015

Keywords

Comments

Conjecture: (i) Any positive rational number r can be written as m/n, where m and n are positive integers such that (prime(prime(prime(m))) + prime(prime(prime(n))))/2 = prime(prime(p)) for some prime p.
(ii) Let p(1,n) = prime(n), and p(m+1,n) = p(m,prime(n)) for m,n = 1,2,3,.... Then, for any integers m > 0 and k > 2, the sequence p(m,n) (n = 1,2,3,...) contains infinitely many nontrivial k-term arithmetic progressions.
(iii) Let m be any positive integer. Then the sequence p(m,n)^(1/n) (n = 1,2,3,...) is strictly decreasing. Also, for any relatively prime integers q > 0 and r, there are infinitely many n > 0 such that p(m,n) == r (mod q).
Note that part (ii) of the conjecture extends the Green-Tao theorem and the third part of the conjecture in A261462. Also, part (iii) in the case m = 1 reduces to the Firoozbakht conjecture on primes and Dirichlet's theorem on primes in arithmetic progressions.

Examples

			a(2) = 755 since prime(prime(prime(755))) + prime(prime(prime(755*2))) = prime(prime(5741)) + prime(prime(12641)) = prime(56611) + prime(135671) = 700897 + 1808581 = 2*1254739 = 2*prime(96797) = 2*prime(prime(9319)) with 9319 prime.

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, A006450, A261437, A261462.

Programs

  • Mathematica
    f[n_]:=Prime[Prime[Prime[n]]]
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]&&PrimeQ[PrimePi[PrimePi[p]]]
Do[k=0;Label[bb];k=k+1;If[PQ[(f[k]+f[k*n])/2],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,60}]

A261625 Number of primes p <= n such that (p-1)*n+1 is prime.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 2, 1, 2, 3, 2, 2, 4, 1, 4, 3, 1, 5, 2, 2, 5, 4, 3, 3, 4, 3, 5, 6, 3, 5, 3, 2, 6, 5, 5, 5, 3, 2, 5, 6, 3, 4, 6, 2, 7, 9, 2, 5, 5, 3, 9, 7, 1, 5, 7, 5, 5, 8, 2, 8, 7, 3, 8, 7, 5, 7, 6, 3, 6, 9, 5, 9, 7, 4, 6, 8, 3, 8, 9, 3
Offset: 1

Views

Author

Zhi-Wei Sun, Aug 27 2015

Keywords

Comments

Conjecture: a(n) > 0 for all n > 1.

Examples

			a(53) = 1 since 3 and (3-1)*53+1 = 107 are both prime.

Links

Crossrefs

Cf. A000040, A034693, A258803, A258811, A261437.

Programs

  • Mathematica
    Do[r=0;Do[If[PrimeQ[(Prime[k]-1)n+1],r=r+1],{k,1,PrimePi[n]}];Print[n," ",r];Continue,{n,1,80}]
  • PARI
    a(n) = my(nb=0); forprime(p=2, n, if (isprime((p-1)*n+1), nb++)); nb; \\ Michel Marcus, Aug 27 2015
Showing 1-2 of 2 results.

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

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