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.

A261352 Primes p such that prime(p)+2 = prime(q)*prime(r) for distinct primes q and r.

Original entry on oeis.org

11, 23, 167, 197, 223, 317, 359, 461, 593, 619, 859, 1283, 1289, 1327, 1487, 1759, 1879, 2557, 2579, 2749, 2879, 3617, 4159, 4783, 5081, 5333, 5531, 5689, 5783, 5867, 6427, 6521, 7589, 7681, 7727, 7753, 9041, 9157, 9283, 9479, 10111, 10289, 10853, 11261, 11779, 11867, 12541, 13309, 13399, 13687
Offset: 1

Views

Author

Zhi-Wei Sun, Aug 15 2015

Keywords

Comments

Conjecture: The sequence has infinitely many terms.
See also A261354 for a similar conjecture, and A261353 for a stronger conjecture.
Recall that a prime p is called a Chen prime if p+2 is a product of at most two primes. It is known that there are infinitely many Chen primes.

Examples

			a(1) = 11 since 11 is a prime, and prime(11)+2 = 3*11 = prime(2)*prime(5) with 2 and 5 both prime.
a(2) = 23 since 23 is a prime, and prime(23)+2 = 5*17 = prime(3)*prime(7) with 3 and 7 both prime.

References

  • Jing-run Chen, On the representation of a large even integer as the sum of a prime and a product of at most two primes, Sci. Sinica 16(1973), 157-176.
  • 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, A109611, A261282, A261353, A261354, A261361.

Programs

  • Mathematica
    Dv[n_]:=Divisors[n]
PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
q[n_]:=Length[Dv[n]]==4&&PQ[Part[Dv[n],2]]&&PQ[Part[Dv[n],3]]
f[k_]:=Prime[Prime[k]]+2
n=0;Do[If[q[f[k]],n=n+1;Print[n," ",Prime[k]]],{k,1,1620}]

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}]

A261385 Least positive integer k such that (prime(prime(k))-1)*(prime(prime(k*n))-1) = prime(p)-1 for some prime p.

Original entry on oeis.org

1, 3, 221, 15, 13, 137, 63, 103, 44, 2, 31, 3, 45, 3, 4, 104, 38, 237, 61, 19, 56, 183, 22, 11, 15, 374, 9, 5, 42, 97, 2, 47, 4, 19, 23, 399, 3, 103, 29, 10, 2, 109, 51, 1, 52, 80, 23, 64, 76, 2, 218, 3, 7, 98, 4, 145, 10, 12, 213, 87, 36, 181, 28, 169, 71, 25, 72, 71, 54, 50
Offset: 1

Views

Author

Zhi-Wei Sun, Aug 17 2015

Keywords

Comments

Conjecture: Let d be any nonzero integer. Then each positive rational number r can be written as m/n, where m and n are positive integers with (prime(prime(m))+d)*(prime(prime(n))+d) = prime(p)+d for some prime p.
This conjecture implies that for any nonzero integer d the equation x*y = z with x,y,z in the set {prime(p)+d: p is prime} has infinitely many solutions.

Examples

			a(3) = 221 since (prime(prime(221))-1)*(prime(prime(221*3))-1) = (prime(1381)-1)*(prime(4957)-1) = 11446*48130 = 550895980 = prime(28890079)-1 with 28890079 prime.

Links

Crossrefs

Cf. A000040, A257938, A261353.

Programs

  • Mathematica
    f[n_]:=Prime[Prime[n]]-1
PQ[p_]:=PrimeQ[p]&&PrimeQ[PrimePi[p]]
Do[k=0;Label[bb];k=k+1;If[PQ[f[k]*f[k*n]+1],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,70}]
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.