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.

A086361 a(n)=A085956(3n).

Original entry on oeis.org

13, 61, 127, 73, 61, 397, 211, 97, 163, 181, 463, 5689, 547, 1093, 271, 673, 1123, 10909, 229, 241, 4663, 661, 967, 1873, 7951, 1093, 16363, 5209, 349, 541, 373, 7873, 397, 409, 421, 433, 4219, 9349, 15679, 1201, 1723, 7309, 15739, 3433, 811, 1933
Offset: 1

Views

Author

Labos Elemer, Jul 22 2003

Keywords

Examples

			n=144:3n=432,6n=864, a(144)=120097 prime with (120097-1)/864=139 prime and with 864.12097+1=103763809 prime.

Crossrefs

Cf. A085956, A086362.

Programs

  • PARI
    a(n) = {n = 3*n; p = 2; while (!(isprime(2*n*p+1) && (type(q=(p-1)/(2*n))== "t_INT") && isprime(q)), p = nextprime(p+1)); p;} \\ Michel Marcus, Feb 25 2014

A086362 a(n) = A085956(3n+1).

Original entry on oeis.org

5, 17, 239, 41, 131, 1889, 419, 89, 101, 113, 2543, 2789, 149, 881, 173, 9293, 491, 14249, 3191, 1973, 3539, 21377, 7103, 281, 5987, 38153, 317, 2789, 6971, 353, 214943, 42677, 3299, 11801, 2267, 27773, 29867, 10529, 461, 1181, 2663, 129209
Offset: 0

Views

Author

Labos Elemer, Jul 22 2003

Keywords

Comments

A086361 and A086362 includes most probably the solvable [or small?] cases of A085956, where n=3k-1 has probably only two solutions, at [k,n,a(n)]=[1,2,13] and at [2,5,31].

Examples

			n=50:3n+1=151,6n+2=302, a(50)=93923 prime with (93922-1)/302=311, prime and with 302.93923+1=28364747 prime; so the {311,93923,28364747} "generalized [short] Cunningham-chain" is defined.

Links

Crossrefs

Cf. A085956, A086361.

Programs

  • Mathematica
    Table[Function[m, p = 2; While[Nand @@ PrimeQ@ {2 m p + 1, (p - 1)/(2 m)}, p = NextPrime@ p]; p][3 n + 1], {n, 0, 41}] (* Michael De Vlieger, May 15 2017 *)

A023287 Primes that remain prime through 3 iterations of function f(x) = 6x + 1.

Original entry on oeis.org

61, 101, 1811, 3491, 4091, 5711, 5801, 6361, 7121, 10391, 10771, 11311, 13421, 15131, 17791, 18911, 19471, 20011, 24391, 25601, 25951, 30091, 35251, 41911, 45631, 47431, 55631, 58711, 62921, 67891, 70451, 70571, 72271, 74051, 74161, 75431, 80471, 86341
Offset: 1

Views

Author

David W. Wilson

Keywords

Comments

Primes p such that s1=p, s2=6*s1+1, s3=6*s2+1 and s4=6*s3+1 are primes forming a special chain of four primes. A fifth term in such a chain cannot arise. See A085956, A086361, A086362.
Entries in chains are congruent to {1,7,3,9} mod 10.

Examples

			First chain is {61, 367, 2203, 13219};
319th chain is {1291391, 7748347, 46490083, 278940499}.

Links

Crossrefs

Cf. A085956, A086361, A086362, A023330, A059766.
Subsequence of A007693, A023256, and A024899.

Programs

  • Magma
    [n: n in [1..150000] | IsPrime(n) and IsPrime(6*n+1) and IsPrime(36*n+7) and IsPrime(216*n+43)] // Vincenzo Librandi, Aug 04 2010
  • Mathematica
    k=0; m=6; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; If[PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3], k=k+1; Print[{k, n, s, s1, s2, s3}]], {n, 1, 100000}] (* edited by Zak Seidov, Feb 08 2011 *)
thrQ[n_]:=AllTrue[Rest[NestList[6#+1&,n ,3]],PrimeQ]; Select[Prime[Range[9000]],thrQ] (* Harvey P. Dale, Mar 03 2024 *)

Formula

{p, 6p+1, 36p+7, 216p+43} are all primes, where p is prime.

Extensions

Additional comments from Labos Elemer, Jul 23 2003

A085958 Primes p such that 8p +1 and (p-1)/8 are primes.

Original entry on oeis.org

17, 137, 809, 857, 3209, 6569, 7577, 9209, 11897, 16217, 17657, 19289, 21929, 23417, 30809, 35129, 37529, 41849, 46889, 53609, 56009, 66377, 70457, 74729, 76697, 78809, 80537, 88937, 91577, 95177, 96857, 103289, 129497, 141257, 165449, 166169
Offset: 1

Views

Author

Amarnath Murthy, Jul 16 2003

Keywords

Examples

			137 is a term as (137-1)/8 = 17 and 137*8 +1 = 1097 both are primes.

Crossrefs

Cf. A085956, A085957.

Programs

  • Mathematica
    Prime[ Select[ Range[15208], PrimeQ[ 8Prime[ # ] + 1] && PrimeQ[(Prime[ # ] - 1)/8] & ]]
Select[Prime[Range[16000]],PrimeQ[8#+1]&&PrimeQ[(#-1)/8]&] (* Harvey P. Dale, Jan 23 2013 *)

Extensions

Edited and extended by Zak Seidov and Robert G. Wilson v, Jul 18 2003

A085957 Primes p such that 6p + 1 and (p-1)/6 are primes.

Original entry on oeis.org

13, 103, 283, 367, 607, 787, 907, 1987, 2203, 2767, 3643, 4507, 5623, 5827, 6907, 7927, 7963, 8167, 8887, 9067, 9643, 10867, 11083, 11443, 12487, 13723, 14083, 14143, 16747, 17107, 17827, 19507, 19543, 20743, 20947, 21487, 22567, 24043
Offset: 1

Views

Author

Amarnath Murthy, Jul 16 2003

Keywords

Crossrefs

Cf. A085956, A085958.

Programs

  • Mathematica
    Prime[ Select[ Range[2738], PrimeQ[ 6Prime[ # ] + 1] && PrimeQ[(Prime[ # ] - 1)/6] & ]]

Extensions

Edited and extended by Zak Seidov and Robert G. Wilson v, Jul 18 2003

A023347 Primes which remain prime through 5 iterations of function f(x) = 8x + 1.

Original entry on oeis.org

831167, 1154567, 2502767, 3019787, 3675197, 5056577, 6352487, 14519177, 26724377, 43003577, 47378927, 47695607, 56406197, 86332457, 86611757, 99568757, 121967987, 126435527, 127990997, 128149127, 128975057, 145281557, 155715407
Offset: 1

Views

Author

David W. Wilson

Keywords

Examples

			First chain is {831167, 6649337, 53194697, 425557577, 3404460617, 27235684937};
If p is congruent to {1,3,7,9} mod 10, then consecutive iterates are congruent to {9,5,7,3}, {3,1,7,5}, {5,9,7,1} respectively; so only 10k+7 may remain prime through five iterations, as sequence demonstrates nicely. - _Labos Elemer_, Jul 23 2003

Links

Crossrefs

Cf. A085956, A086361, A086362, A023330, A059766, A023287, A000668, A076481, A086127.
Subsequence of A005123, A023228, A023260, A023291, and A023319.

Programs

  • Mathematica
    k=0; m=8; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; s4=m*s3+1; s5=m*s4+1; If[PrimeQ[s]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5], k=k+1; Print[s]], {n, 1, 1000000}]
it5Q[n_]:=AllTrue[Rest[NestList[8#+1&,n,5]],PrimeQ]; Select[Prime[Range[ 9*10^6]],it5Q] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 12 2014 *)

Formula

{p, 8p+1, 64p+9, 512p+73, 4096p+585, 32768p+4681} are all primes, where the initial p is prime.
a(n) == 197 (mod 210). - John Cerkan, Nov 04 2016

A086127 Numbers k such that k remains prime after five iteration of function f(j) = 14*f(j)+1, starting at f(1) = prime.

Original entry on oeis.org

4889, 18059, 62639, 225527, 557093, 604973, 700703, 804077, 806903, 837077, 1341203, 1363403, 1932197, 2004269, 2062703, 2284637, 2797463, 3157379, 3493103, 3746399, 3995687, 4155413, 4227893, 4493297, 5534939, 5708603
Offset: 1

Views

Author

Labos Elemer, Jul 23 2003

Keywords

Comments

{p, 14p+1, 196p+15, 2744p+211, 38416p+2955, 537824p+41371} are all primes, where p is prime.

Examples

			First chain is: {4889,68447,958259,13415627,187818779,2629462907}.
10th chain is {837077,11719079,164067107,2296939499,32157152987,450200141819}.

Crossrefs

Cf. A085956, A086361, A086362, A023330, A059766, A023287, A000668, A076481.

Programs

  • Mathematica
    k=0; m=14; Do[s=Prime[n]; s1=m*s+1; s2=m*s1+1; s3=m*s2+1; s4=m*s3+1; s5=m*s4+1; If[PrimeQ[s]&&PrimeQ[s1]&&PrimeQ[s2]&&PrimeQ[s3]&&PrimeQ[s4]&&PrimeQ[s5], k=k+1; Print[s]], {n, 1, 1000000}]
Select[Range[6000000],And@@PrimeQ[NestList[14#+1&,#,5]]&] (* Harvey P. Dale, Sep 17 2012 *)
Showing 1-7 of 7 results.

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