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 1-7 of 7 results.

A259492 Least positive integer k such that prime(k)-k, prime(k)+k, prime(k*n)-k*n, prime(k*n)+k*n, prime(k)+k*n and prime(k*n)+k are all prime.

Original entry on oeis.org

4, 48852, 6, 27330, 89814, 13080, 9570, 44592, 6762, 28560, 1560, 8580, 2958, 672, 9816, 6300, 40050, 53580, 3354, 858, 4530, 100650, 182520, 49740, 48660, 25296, 66990, 87120, 43680, 6840, 52122, 2970, 22770, 15888, 34704, 406350, 67890, 99630, 92490, 83064, 28614, 8580, 32070, 42, 50442, 38676, 818202, 30450, 47880, 4620
Offset: 1

Views

Author

Zhi-Wei Sun, Jun 28 2015

Keywords

Comments

Conjecture: Any positive rational number r can be written as m/n with prime(m)-m, prime(m)+m, prime(n)-n, prime(n)+n, prime(m)+n and m+prime(n) all prime.

Examples

			a(3) = 6 since prime(6)-6 = 7, prime(6)+6 = 19, prime(6*3)-6*3 = 43, prime(6*3)+6*3 = 79, prime(6)+6*3 = 31 and prime(6*3)+6 = 67 are all 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, A258836, A259487, A259540.

Programs

  • Mathematica
    PQ[k_]:=PrimeQ[Prime[k]-k]&&PrimeQ[Prime[k]+k]
QQ[m_,n_]:=PQ[m]&&PQ[n]&&PrimeQ[Prime[m]+n]&&PrimeQ[m+Prime[n]]
Do[k=0;Label[bb];k=k+1;If[QQ[k,n*k], Goto[aa], Goto[bb]]; Label[aa]; Print[n, " ", k];Continue,{n,1,50}]

A259539 Numbers m with m-1, m+1 and prime(m)+2 all prime.

Original entry on oeis.org

60, 828, 858, 1032, 1050, 1230, 1320, 1878, 2028, 2340, 3252, 3390, 3462, 4548, 5502, 6870, 6948, 7590, 7878, 8010, 9438, 9720, 9858, 10038, 10068, 10302, 11490, 11718, 13932, 14388, 15138, 15270, 15288, 16068, 16188, 16230, 17208, 17292, 17838, 17910
Offset: 1

Views

Author

Zhi-Wei Sun, Jun 30 2015

Keywords

Comments

Conjecture: The sequence contains infinitely many terms.
This is stronger than the Twin Prime Conjecture, and weaker than the conjecture in A259540.
Subsequence of A014574.

Examples

			a(1) = 60 since 60-1 = 59, 60+1 = 61 and prime(60)+2 = 283 are all 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, A014574, A236458, A259540.

Programs

  • Mathematica
    n=0;Do[If[PrimeQ[k-1]&&PrimeQ[k+1]&&PrimeQ[Prime[k]+2],n=n+1;Print[n," ",k]],{k,1,18000}]
  • PARI
    lista(nn) = {forprime(p=2, nn, if (isprime(p+2) && isprime(prime(p+1)+2), print1(p+1, ", ")));} \\ Michel Marcus, Jun 30 2015

A259531 Least positive integer k such that p(k)^2 + p(k*n)^2 is prime, where p(.) is the partition function given by A000041, or 0 if no such k exists.

Original entry on oeis.org

1, 1, 14, 11, 6, 31, 2, 34, 2, 76, 1, 100, 71, 38, 1, 7, 62, 1128, 1, 180, 123, 15, 174, 128, 4, 111, 110, 2, 4, 2, 2241, 21, 144, 416, 397, 31, 11, 8, 15, 5, 91, 56, 53, 23, 89, 18, 25, 341, 12, 1, 66, 454, 159, 36, 573, 26, 2, 488, 72, 416, 802, 440, 28, 30, 595, 17, 236, 947, 1289, 1287, 1000, 367, 80, 407, 1, 77, 938, 150, 36, 1
Offset: 1

Views

Author

Zhi-Wei Sun, Jul 02 2015

Keywords

Comments

Conjecture: Any positive rational number r can be written as m/n, where m and n are positive integers with p(m)^2 + p(n)^2 prime.
For example, 4/5 = 124/155, and the number p(124)^2 + p(155)^2 = 2841940500^2 + 66493182097^2 = 4429419891190341567409 is prime.
We also guess that any positive rational number can be written as m/n, where m and n are positive integers with p(m)+p(n) (or p(m)*p(n)-1, or p(m)*p(n)+1) prime.

Examples

			a(5) = 6 since p(6)^2 + p(6*5)^2 = 11^2 + 5604^2 = 31404937 is 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, A000041, A258803, A258836, A259487, A259492, A259540.

Programs

  • Mathematica
    Do[k=0;Label[bb];k=k+1;If[PrimeQ[PartitionsP[k]^2+PartitionsP[k*n]^2],Goto[aa],Goto[bb]];Label[aa];Print[n," ",k]; Continue,{n,1,80}]

A261339 Least positive integer k such that both k and k*n belong to the set {m>0: m+1, m^2+1 and m^2+prime(m)^2 are all prime}.

Original entry on oeis.org

1, 1, 47500, 20440, 2, 124560, 17850, 2730, 185550, 1, 518910, 429180, 10, 687480, 81030, 36, 1568340, 2, 1165750, 7410, 10, 6780, 481140, 10, 10, 5430, 240, 2730, 72660, 2080, 18700, 291720, 295080, 52860, 5430, 1, 81030, 56400, 12490, 43590, 124560, 40030, 5170, 278700, 2091850, 131320, 184110, 11206510, 12910, 1245780
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 with m and n in the set {k>0: k+1, k^2+1 and k^2+prime(k)^2 are all prime}.
For example, 5/8 = 3567600/5708160 with 3567600+1, 3567600^2+1 = 12727769760001, 3567600^2 + prime(3567600)^2 = 3567600^2 + 60098671^2 = 3624578025726241, 5708160+1, 5708160^2+1 = 32583090585601, and 5708160^2 + prime(5708160)^2 = 5708160^2 + 99018553^2 = 9837256928799409 all prime.
The conjecture implies that there are infinitely many primes p with (p-1)^2+1 and (p-1)^2+prime(p-1)^2 both prime.
We also guess that any positive rational number can be written as m/n, where m and n are positive integers with m^2+prime(m)^2, m^2+prime(n)^2, n^2+prime(m)^2 and n^2+prime(n)^2 all prime.

Examples

			a(3) = 47500 since 47501, 47500^2 + 1 = 2256250001, 47500^2 + prime(47500)^2 = 47500^2 + 578827^2 = 337296945929, 47500*3 + 1 = 142501, (47500*3)^2 + 1 = 20306250001, and (47500*3)^2 + prime(47500*3)^2 = 142500^2 + 1907023^2 = 3657042972529 are all 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, A005574, A236193, A259492, A259540, A260120.

Programs

  • Mathematica
    PQ[n_]:=PrimeQ[n+1]&&PrimeQ[n^2+1]&&PrimeQ[n^2+Prime[n]^2]
Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,50}]

A261528 Least positive integer k such that both k and k*n belong to the set {m>0: prime(m)+2 is prime with prime(prime(m)+2) = prime(prime(m))+6}.

Original entry on oeis.org

2, 891, 81002, 814812, 86050, 5917, 65527, 109853, 2563344, 25379, 2640232, 266076, 775889, 67387, 68111, 37950, 353416, 347139, 56390, 11299, 89491, 545458, 910786, 353416, 1913477, 9025, 111569, 511796, 1456228, 37909, 1494675, 212092, 69352, 107769, 300657, 1155675, 391972, 1073031, 55074, 49892
Offset: 1

Views

Author

Zhi-Wei Sun, Aug 23 2015

Keywords

Comments

Conjecture: Any positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+2 is prime with prime(prime(k)+2) = prime(prime(k))+6}.
This implies that there are infinitely many twin prime pairs {p, p+2} with prime(p+2) - prime(p) = 6.
Note that if prime(n+2)-prime(n) = 6 then prime(n+1)-prime(n) = 2 or 4.

Examples

			a(1) = 2 since 2*1 = 2, and prime(2)+2 = 3+2 = 5 is prime with prime(5)-prime(3) = 11-5 = 6.
a(2) = 891 since prime(891)+2 = 6947 + 2 = 6949 is prime with prime(6949)-prime(6947) = 70123-70117 = 6, and prime(891*2)+2 = 15269 + 2 = 15271 is prime with prime(15271)-prime(15269) = 167119-167113 = 6.

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, A023201, A046117, A259487, A259540, A261533.

Programs

  • Mathematica
    f[n_]:=Prime[n]
PQ[k_]:=PrimeQ[f[k]+2]&&f[f[k]+2]-f[f[k]]==6
Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,40}]

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}]

A261541 Least positive integer m such that both m and m*n belong to the set {k>0: prime(k)+2, prime(k)+6, prime(k)+8 are all prime}.

Original entry on oeis.org

3, 358712, 34772, 79631, 1822685, 22865, 2066, 2593722, 26, 3418900, 26, 711611, 286, 1493190, 882854, 513312, 1707237, 788232, 913695, 1980985, 7147, 443152, 479580, 2589105, 865432, 265243, 103641, 160536, 398360, 851672
Offset: 1

Views

Author

Zhi-Wei Sun, Aug 24 2015

Keywords

Comments

Conjecture: (i) Each positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+2, prime(k)+6 and prime(k)+8 are all prime}.
(ii) Any positive rational number r can be written as m/n with m and n in the set {k>0: prime(k)+4, prime(k)+6 and prime(k)+10 are all prime}.
For example, 3/4 = 20723892/27631856, and prime(20723892)+2 = 387875561+2 = 387875563, prime(20723892)+6 = 387875567, prime(20723892)+8 = 387875569, prime(27631856)+2 = 525608591+2 =525608593, prime(27631856)+6 = 525608597, prime(27631856)+8 = 525608599 are all prime. Also, 3/4 = 599478/799304, and prime(599478)+4 = 8951857+4 = 8951861, prime(599478)+6 = 8951863, prime(599478)+10 = 8951867, prime(799304)+4 = 12183943+4 = 12183947, prime(799304)+6 = 12183949, prime(799304)+10 = 12183953 are all prime.
Part (i) of the conjecture implies that there are infinitely many primes p with p+2, p+6 and p+8 all prime, while part (ii) implies that there are infinitely many primes p with p+4, p+6 and p+10 all prime.

Examples

			a(1) = 3 since 3*1 = 3, and prime(3)+2 = 5+2 =7, prime(3)+6 = 11 and prime(3)+8 = 13 are all prime.
a(2) = 358712 since prime(358712)+2 = 5158031+2 = 5158033, prime(358712)+6 = 5158037, prime(358712)+8 = 5158039, prime(358712*2)+2 = 10852601+2 = 10852603, prime(358712*2)+6 = 10852607 and prime(358712*2)+8 = 10852609 are all 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, A007530, A052378, A236511, A259487, A259540.

Programs

  • Mathematica
    f[n_]:=Prime[n]
PQ[k_]:=PrimeQ[f[k]+2]&&PrimeQ[f[k]+6]&&PrimeQ[f[k]+8]
Do[k=0;Label[bb];k=k+1;If[PQ[k]&&PQ[k*n],Goto[aa],Goto[bb]];Label[aa];Print[n," ", k];Continue,{n,1,30}]
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