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 1-3 of 3 results.

A261361 Primes p such that 2*prime(p) + 1 = prime(q) for some prime q.

Original entry on oeis.org

3, 13, 173, 463, 523, 823, 971, 991, 1291, 1543, 2113, 4003, 4019, 4649, 5923, 6037, 6101, 7649, 10103, 12539, 12841, 17203, 17569, 18013, 21193, 22093, 23321, 25111, 27197, 31231, 32009, 32117, 33349, 34687, 35423, 35449, 37747, 39619, 41729, 41759, 42017, 43237, 43331, 44741, 45841, 50021, 51437, 52489, 55921, 56891
Offset: 1

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Author

Zhi-Wei Sun, Aug 16 2015

Keywords

Comments

Conjecture: The sequence contains infinitely many terms. In general, if a,b,c are positive integers with gcd(a,b) = gcd(a,c) = gcd(b,c) = 1, and a+b+c is even and a is not equal to b, then there are infinitely many prime pairs {p,q} such that a*prime(p) - b*prime(q) = c.
See also A261362 for a stronger conjecture.
Recall that a prime p is called a Sophie Germain prime if 2*p+1 is also prime. A well-known conjecture states that there are infinitely many Sophie Germain primes.

Examples

			a(1) = 3 since 3 is a prime, and 2*prime(3)+1 = 2*5+1 = 11 = prime(5) with 5 prime.
a(3) = 173 since 173 is a prime, and 2*prime(173)+1 = 2*1031+1 = 2063 = prime(311) with 311 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, A005384, A260886, A260888, A261352, A261354, A261362.

Programs

  • Mathematica
    f[n_]:=2*Prime[Prime[n]]+1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
n=0;Do[If[PQ[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,5800}]

A261362 Least positive integer k such that both k and k*n belong to the set {m>0: 2*prime(prime(m))+1 = prime(p) for some prime p}.

Original entry on oeis.org

2, 21531, 2, 35434, 11107, 35175, 24674, 64624, 127943, 1981, 155709, 50657, 74313, 11479, 6, 1981, 43405, 40859, 74229, 2, 154292, 51711, 29460, 29011, 42001, 28352, 2979, 85836, 6936, 186608, 3705, 14402, 25525, 96192, 6, 113433, 164, 787, 71873, 3365, 93169, 47219, 43128, 184740, 2, 78329, 13656, 6936, 139469, 26713
Offset: 1

Views

Author

Zhi-Wei Sun, Aug 16 2015

Keywords

Comments

Conjecture: Let a,b,c be positive integers with gcd(a,b) = gcd(a,c) = gcd(b,c) = 1. If a+b+c is even and a is not equal to b, then any positive rational number r can be written as m/n with m and n in the set {k>0: a*prime(p) - b*prime(prime(k)) = c for some prime p}.
This implies the conjecture in A261361.

Examples

			a(2) = 21531 since 2*prime(prime(21531))+1 = 2*prime(243799)+1 = 2*3403703+1 = 6807407 = prime(464351) with 464351 prime, and 2*prime(prime(21531*2))+1 = 2*prime(520019)+1 = 2*7686083+1 = 15372167 = prime(993197) with 993197 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, A005384, A260886, A261353, A261354, A261361.

Programs

  • Mathematica
    f[n_]:=2*Prime[Prime[n]]+1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0;Label[bb];k=k+1;If[PQ[f[k]]&&PQ[f[k*n]],Goto[aa],Goto[bb]];Label[aa];Print[n," ", 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 1-3 of 3 results.

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