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}]

A261353 Least positive integer k such that prime(prime(k))*prime(prime(k*n)) = prime(p)+2 for some prime p.

Original entry on oeis.org

11, 2, 1, 606, 350, 166, 53, 1865, 7, 45, 1308, 68, 215, 61, 256, 13, 248, 90, 1, 1779, 796, 1, 4, 444, 650, 55, 157, 303, 82, 84, 25, 3, 1912, 621, 128, 205, 164, 1091, 61, 12, 337, 1, 303, 15, 23, 418, 212, 23, 2494, 1, 472, 771, 1, 36, 8, 46, 8, 18, 264, 22, 725, 85, 65, 231, 606, 3, 1, 43, 144, 164
Offset: 1

Views

Author

Zhi-Wei Sun, Aug 15 2015

Keywords

Comments

Conjecture: a(n) exists for any n > 0. In general, any positive rational number r can be written as m/n, where m and n are positive integers such that prime(prime(m))*prime(prime(n)) = prime(p)+2 for some prime p.
This implies that the sequence A261352 has infinitely many terms.

Examples

			a(1) = 11 since prime(prime(11))*prime(prime(11*1)) = prime(31)^2 = 127^2 = 16129 = prime(1877)+2 with 1877 prime.
a(4) = 606 since prime(prime(606))*prime(prime(606*4)) = prime(4457)*prime(21589) = 42643*244471 = 10424976853 = prime(473490161)+2 with 473490161 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, A261352.

Programs

  • Mathematica
    f[n_]:=Prime[Prime[n]]
PQ[p_]:=PrimeQ[p]&&PrimeQ[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,1,70}]

A261295 Least positive integer k such that both k and k*n belong to the set {m>0: prime(m) = prime(p)+2 for some prime p}.

Original entry on oeis.org

3, 3, 6, 578, 18, 3, 6, 90, 1868, 374, 4, 674, 278, 3, 6, 114, 3534, 110, 6, 354, 4, 14, 28464, 2790, 84, 4452, 2802, 3, 6, 3, 90, 2820, 354, 110, 4080, 278, 44, 3, 2712, 18, 3012, 90, 14, 12672, 44, 14, 1572, 1124, 720, 42, 114, 44, 84, 2790, 42, 90, 42, 3, 6, 84, 44, 1572, 3068, 1742, 2394, 174, 110, 744, 3020, 578
Offset: 1

Views

Author

Zhi-Wei Sun, Aug 14 2015

Keywords

Comments

Conjecture: (i) Any positive rational number r can be written as m/n with m and n in the set S = {k>0: prime(k) = prime(p)+2 for some prime p} = {p+1: p and prime(p)+2 are both prime}.
(ii) Any positive rational number r can be written as m/n with m and n in the set T = {k>0: prime(k) = prime(p)-2 for some prime p} = {p-1: p and prime(p)-2 are both prime}.
(iii) Any positive rational number r not equal to 1 can be written as m/n with m in S and n in T, where the sets S and T are given in parts (i) and (ii).
For example, 4/5 = 15648/19560 with 15647, prime(15647)+2 = 171763, 19559 and prime(19559)+2 = 219409 all prime; and 4/5 = 67536/84420 with 67537, prime(67537)-2 = 848849, 84421 and prime(84421)-2 = 1081937 all prime. Also, 4/5 = 8/10 with 7, prime(7)+2 = 19, 11 and prime(11)-2 = 29 all prime; and 5/4 = 8220/6576 with 8221, prime(8221)+2 = 84349, 6577 and prime(6577)-2 = 65837 all prime.

Examples

			a(3) = 6 since prime(6) = 13 = prime(5)+2 with 5 prime, and prime(6*3) = 61 = prime(17)+2 with 17 prime.
a(4) = 578 since prime(578) = 4219 = prime(577)+2 with 577 prime, and prime(578*4) = 20479 = prime(2311)+2 with 2311 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, A001359, A006512, A218829, A258803, A259487, A259540, A261281, A261282.

Programs

  • Mathematica
    f[n_]:=Prime[n]
PQ[n_]:=PrimeQ[n]&&PrimeQ[PrimePi[n]]
Do[k=0;Label[bb];k=k+1;If[PQ[f[k]-2]&&PQ[f[k*n]-2],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,70}]
Showing 1-3 of 3 results.

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