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.

A239874 Integers k such that 2*k^2 + 1 and 2*k^3 + 1 are prime.

Original entry on oeis.org

1, 6, 9, 21, 27, 30, 72, 96, 99, 162, 186, 204, 237, 264, 297, 321, 357, 360, 375, 492, 537, 621, 759, 819, 834, 897, 936, 1065, 1242, 1326, 1329, 1359, 1419, 1494, 1506, 1596, 1662, 1704, 1740, 1749, 1761, 1842, 1869, 2157, 2175, 2250, 2274, 2451, 2547
Offset: 1

Views

Author

Zak Seidov, Mar 28 2014

Keywords

Comments

All terms > 1 are multiples of 3. Also, no term is congruent to 3 modulo 5.

Links

Crossrefs

Intersection of A089001 and A168550.

Programs

  • Maple
    select(t -> isprime(2*t^2+1) and isprime(2*t^3+1), [$1..6000]); # Robert Israel, Nov 03 2024
  • Mathematica
    s={1};Do[If[PrimeQ [2k^2+1]&&PrimeQ[2k^3+1],AppendTo[s,k]],{k,3,10^3,3}];s
Select[Range[3500], PrimeQ[2 #^2 + 1] && PrimeQ[2 #^3 + 1]&] (* Vincenzo Librandi, Mar 29 2014 *)
  • PARI
    s=[]; for(n=1, 4000, if(isprime(2*n^2+1) && isprime(2*n^3+1), s=concat(s, n))); s \\ Colin Barker, Mar 28 2014

A201107 Primes of the form 2k^3+1.

Original entry on oeis.org

3, 17, 251, 433, 1459, 2663, 3457, 16001, 18523, 35153, 39367, 48779, 54001, 65537, 85751, 170369, 370387, 410759, 432001, 715823, 746497, 913067, 1272113, 1557377, 1714751, 1769473, 1940599, 2450087, 2735263, 3456001, 4812209, 5488001, 6615899, 6750001
Offset: 1

Views

Author

Vincenzo Librandi, Nov 27 2011

Keywords

Links

Crossrefs

Cf. A168550, A033562.

Programs

  • Magma
    [a: n in [0..200] | IsPrime(a) where a is 2*n^3+1];
  • Mathematica
    Select[Table[2n^3+1,{n,0,4000}],PrimeQ]
  • Python
    from sympy import isprime
print(list(filter(isprime, (2*k**3+1 for k in range(152))))) # Michael S. Branicky, Jun 17 2021

Formula

a(n) = 2*A168550(n)^3+1. - R. J. Mathar, Aug 20 2019

A239925 Integers n such that 2n^2+1, 2n^3+1, 2n^4+1 and 2n^5+1 are prime.

Original entry on oeis.org

1, 30, 8025, 44250, 49335, 49599, 155061, 218196, 255975, 293754, 324684, 333405, 336045, 367839, 381804, 416796, 476814, 514005, 529650, 558291, 668856, 682716, 747810, 893190, 930336, 933576, 1004004, 1246266, 1270860, 1383126, 1392111, 1427211, 1491645, 1497024, 1745904, 1786551
Offset: 1

Views

Author

Zak Seidov, Mar 29 2014

Keywords

Links

Crossrefs

Subsequence of A239920. Cf. A089001, A168550, A239874.

Programs

  • Mathematica
    Select[Range[0, 2000000], PrimeQ[2 #^2 + 1] && PrimeQ[2 #^3 + 1] && PrimeQ[2 #^4 + 1] && PrimeQ[2 #^5 + 1] &] (* Vincenzo Librandi, Mar 29 2014 *)
Select[Range[179*10^4], AllTrue[2 #^Range[2, 5] + 1, PrimeQ] &] (* Harvey P. Dale, Sep 24 2021 *)
  • PARI
    s=[]; for(n=1, 2000000, if(isprime(2*n^2+1) && isprime(2*n^3+1) && isprime(2*n^4+1) && isprime(2*n^5+1), s=concat(s, n))); s \\ Colin Barker, Mar 29 2014

A239920 Integers n such that 2n^2+1, 2n^3+1 and 2n^4+1 are prime.

Original entry on oeis.org

1, 6, 21, 30, 96, 297, 375, 621, 1359, 1704, 1749, 1761, 3696, 3849, 4467, 8025, 8646, 9834, 11352, 15630, 17397, 17949, 19575, 20274, 27087, 28452, 30504, 32154, 32307, 33666, 35670, 36240, 37785, 37962, 39927, 40617, 42987, 44250, 47559, 49335, 49599
Offset: 1

Views

Author

Zak Seidov, Mar 29 2014

Keywords

Links

Crossrefs

Subsequence of A239874. Cf. A089001, A168550.

Programs

  • Mathematica
    Select[Range[0, 50000], PrimeQ[2 #^2 + 1] && PrimeQ[2  #^3 + 1] && PrimeQ[2 #^4 + 1]&] (* Vincenzo Librandi, Mar 30 2014 *)
  • PARI
    s=[]; for(n=1, 100000, if(isprime(2*n^2+1) && isprime(2*n^3+1) && isprime(2*n^4+1), s=concat(s, n))); s \\ Colin Barker, Mar 29 2014

A309857 Primes p such that 2*p^3+1 is also prime.

Original entry on oeis.org

2, 5, 11, 29, 59, 71, 107, 149, 191, 197, 227, 269, 431, 479, 491, 857, 941, 1019, 1049, 1217, 1259, 1289, 1451, 1601, 1619, 1667, 1709, 1847, 2081, 2237, 2267, 2447, 2549, 2579, 2699, 2711, 2729, 2861, 2879, 2957, 3041, 3089, 3167, 3191, 3209, 3221, 3407, 3719, 3761, 3779
Offset: 1

Views

Author

R. J. Mathar, Aug 20 2019

Keywords

Comments

All terms == 2 (mod 3). - Robert Israel, Aug 22 2019

Links

Crossrefs

Cf. A177104 (2*p^3-1 prime), A309856.

Programs

  • Maple
    select(t -> isprime(t) and isprime(2*t^3+1), [2, seq(i,i=5..10000,6)]); # Robert Israel, Aug 22 2019

Formula

A000040 INTERSECT A168550.

A240099 Integers n such that 2n^k + 1, for k = 2..6, are prime.

Original entry on oeis.org

1, 44250, 1004004, 3490575, 3517335, 5750115, 10729026, 19193559, 20251770, 25284039, 25552194, 30204801, 33733206, 39015405, 47518809, 52463445, 58370025, 69502971, 72009429, 77086380, 78510156, 83972646, 85955475, 89190969, 90499584, 92246199, 95374005
Offset: 1

Views

Author

Zak Seidov, Apr 01 2014

Keywords

Comments

Note that 2n^7+1 may or may not be prime.
First n>1 such that 2n^k+1, for k=2..7, are prime, is a(4) = 3490575.
First n>1 such that 2n^k+1, for k=2..8, are prime, is 83972646.
Subsequence of A239925: a(2) = 44250 = A239925(4), a(3) = 1004004 = A239925(27).

Crossrefs

Cf. A089001, A168550, A239874, A239925.

Programs

  • Mathematica
    Select[Range[10^8],AllTrue[2#^Range[2,6]+1,PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 18 2015 *)

A240105 Integers m such that 2*m^k + 1, for k = 2..7, are prime.

Original entry on oeis.org

1, 3490575, 83972646, 414180489, 476072025, 1881147720, 3020243916, 3188924769, 3285167214, 3543143220, 6593858205, 8239349955, 10914074124, 14102235060, 15455042889, 16196415300, 16588528539, 16636093485, 17688635511, 17929182270, 18997337436, 19290317670, 19347263739
Offset: 1

Views

Author

Zak Seidov, Apr 01 2014

Keywords

Comments

First m>1 such that 2*m^k+1, for k=2..8, are prime, is a(3) = 83972646.
Subsequence of A240099: a(2) = 3490575 = A240099(4), a(3) = 83972646 = A240099(22).

Crossrefs

Cf. A089001, A168550, A239874, A239925, A240099.

Extensions

More terms from Jinyuan Wang, Jun 12 2025
Showing 1-7 of 7 results.

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