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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A260252 Least prime p such that n = (prime(6*q)-1)/(prime(6*p)-1) for some prime q.

Original entry on oeis.org

2, 18253, 3, 19, 2, 41, 43, 1087, 263, 29, 2, 281, 83, 8941, 613, 827, 7, 1867, 811, 139, 919, 13, 59, 11551, 10303, 10903, 2707, 3019, 1297, 5, 7333, 1609, 541, 701, 2281, 499, 2713, 6691, 41, 79, 1447, 1409, 263, 2129, 641, 2467, 7741, 1229, 523, 6781
Offset: 1

Views

Author

Zhi-Wei Sun, Jul 20 2015

Keywords

Comments

Conjecture: Let n be any positive integer, and let s and t belong to the set {1,-1}. Then each positive rational number r can be written as (prime(p*n)+s)/(prime(q*n)+t) with p and q both prime, unless n > r = 1 and {s,t} = {1,-1}.
This extends the conjecture in A258803.
For example, for n = 8, s = t = -1 and r = 16/11, we have (prime(407249*8)-1) /(prime(286411*8)-1) = 54568320/37515720 = r with 407249 and 286411 both prime. Also, for n = 10, s = -1, t = 1, and r = 23/17, we have (prime(1923029*10)-1)/(prime(1444903*10)+1) = 358404768/264907872 = r with 1923029 and 1444903 both prime.

Examples

			a(1) = 2 since 1 = (prime(6*2)-1)/(prime(6*2)-1) with 2 prime.
a(2) = 18253 since 2 = 2868672/1434336 = (prime(6*34673)-1)/(prime(6*18253)-1) with 18253 and 34673 both 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, A258803, A260232.

Programs

  • Mathematica
    PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/6]
Do[k=0;Label[aa];k=k+1;If[PQ[(Prime[6*Prime[k]]-1)*n+1],Goto[bb],Goto[aa]];Label[bb];Print[n, " ", Prime[k]];Continue,{n,1,50}]

A260888 Least prime p such that 2 + 3*pi(p*n) = 4*pi(q*n) for some prime q, where pi(x) denotes the number of primes not exceeding x.

Original entry on oeis.org

3, 2, 41, 211, 23, 83, 43, 23, 7, 3, 601, 109, 23, 251, 31, 251, 7, 41, 149, 157, 293, 3, 103, 41, 2083, 233, 7, 647, 1877, 7, 1117, 599, 7, 937, 487, 7, 251, 149, 7, 439, 83, 3, 7, 43, 643, 7, 157, 157, 1291, 7
Offset: 1

Views

Author

Zhi-Wei Sun, Aug 02 2015

Keywords

Comments

Conjecture: Let a and b be relatively prime positive integers, and let c be any integer. For any positive integer n, there are primes p and q such that a*pi(p*n) - b*pi(q*n) = c.
In the case c = 0, this reduces to the conjecture in A260232.
For example, for a = 20, b = 19, c = 18 and n = 28, we have 20*pi(4549*28)-19*pi(4813*28) = 20*11931-19*12558 = 18 with 4549 and 4813 both prime.

Examples

			a(5) = 23 since 2+3*pi(23*5) = 2+3*30 = 92 = 4*23 = 4*pi(17*5) with 23 and 17 both prime.

Links

Crossrefs

Cf. A000040, A000720, A260140, A260232, A260886.

Programs

  • Mathematica
    f[k_,n_]:=PrimePi[Prime[k]*n]
Do[k=0;Label[bb];k=k+1;If[Mod[3*f[k,n]+2,4]>0,Goto[bb]];Do[If[(3*f[k,n]+2)/4==f[j,n],Goto[aa]];If[(3*f[k,n]+2)/4
Showing 1-2 of 2 results.

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