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.

A260208 Least prime p such that 2p*n+1 = prime(q*n) for some prime q.

Original entry on oeis.org

2, 3, 2, 107, 271, 3, 3, 523, 17, 191, 73, 2707, 587, 2017, 19, 233, 57193, 7583, 9791, 7, 2111, 1373, 43, 109, 1283, 463, 8179, 25583, 7489, 1733, 9011, 7753, 7853, 887, 10141, 71, 1373, 7927, 509, 1433, 4513, 2399, 4211, 26407, 307, 2843, 58579, 3121, 5519, 38371
Offset: 1

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Author

Zhi-Wei Sun, Jul 19 2015

Keywords

Comments

Conjecture: a(n) exists for any n > 0. In general, if a > 0 is even and b is 1 or -1, then for any positive integer n there are primes p and q such that a*p*n+b = prime(q*n).

Examples

			a(5) = 271 since 2*271*5+1 = 2711 = prime(79*5) with 271 and 79 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, A260197.

Programs

  • Mathematica
    PQ[n_,p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]/n]
Do[k=1;While[!PQ[n,2*Prime[k]*n+1],k=k+1];Print[n, " ", Prime[k]],{n,1,50}]

A260219 Least prime p such that pi(p*n)^2 + 1 = prime(q*n) for some prime q.

Original entry on oeis.org

3, 47, 229, 379, 11, 2687, 1181, 24547, 5, 509, 5, 16619, 877, 22543, 3, 9067, 60337, 10667, 997, 24061, 18329, 56099, 1787, 58757, 108883, 416881, 157141, 11003, 14939, 113167, 101957, 77969, 613, 27947, 52153, 158551, 6197, 36607, 25237, 27179, 330689, 203617, 77419, 708269, 87649, 340381, 267601, 67153, 123377, 21617
Offset: 1

Views

Author

Zhi-Wei Sun, Jul 19 2015

Keywords

Comments

Conjecture: a(n) exists for any n > 0. In general, if a,b,c are integers with a > 0 and gcd(a,b,c) = 1, and a+b or c is odd, and b^2 - 4*a*c is not a square, then there are primes p and q such that a*pi(p*n)^2 + b*pi(p*n) + c = prime(q*n).

Examples

			a(1) = 3 since pi(3*1)^2 + 1 = 5 = prime(3*1) with 3 prime.
a(8) = 24547 since pi(24547*8)^2 + 1 = 17686^2 + 1 = 312794597 = prime(2113417*8) with 24547 and 2113417 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, A000720, A002496, A259731, A260120, A260197.

Programs

  • Mathematica
    PQ[n_, p_] := PrimeQ[p] && PrimeQ[PrimePi[p]/n]; Do[k = 0; Label[bb]; k = k + 1; If[PQ[n, PrimePi[Prime[k] * n]^2 + 1], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", Prime[k]]; Continue,{n, 50}]
  • PARI
    first(m)={my(v=vector(m),p,q,n);for(n=1,m,p=0;while(1,p++;q=1;while(primepi(prime(p)*n)^2 +1 >= prime(prime(q)*n), if(primepi(prime(p)*n)^2 +1 == prime(prime(q)*n),v[n]=prime(p);break(2),q++;))));v;} /* Anders Hellström, Jul 19 2015 */
Showing 1-2 of 2 results.

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