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.

A178336 Smaller member of a twin prime pair of the form (k^3 + 2, k^3 + 4).

Original entry on oeis.org

3, 29, 91127, 250049, 328511, 2146691, 47832149, 121287377, 170953877, 194104541, 693154127, 979146659, 1167575879, 1664006627, 5079577961, 6219352721, 8678316377, 10289109377, 10633486601, 13980103931, 17474794877, 28066748321, 28736971049
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 25 2010

Keywords

Examples

			3 = 1^3+2 = prime(2) and 5 = 1^3+4 = prime(3) are a twin prime pair, so 3 becomes the first term.
91127 = 45^3+2 = prime(8811) and 91129 = 45^3+4 = prime(8812) are a twin prime pair, so 91127 is a term.

References

  • Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, B. G. Teubner, Leipzig u. Berlin, 1909

Links

Crossrefs

Cf. A013159, A053703, A132282, A144953, A173255, A178228.

Programs

  • Mathematica
    Select[Range[3100]^3+2,PrimeQ[#]&&PrimeQ[#+2]&] (* Harvey P. Dale, May 26 2012 *)

Formula

a(n) = A178337(n)^3 + 2.

Extensions

Keyword:base removed, 2 missing terms inserted by R. J. Mathar, Jun 27 2010

A174370 Lesser member p of a twin prime pair (p, p + 2) such that 2p + 3(p + 2) is a perfect square.

Original entry on oeis.org

71, 191, 6551, 9767, 18119, 21647, 27527, 35447, 46271, 79631, 103391, 103967, 121367, 127679, 161639, 207671, 241559, 254927, 264959, 273311, 380327, 421079, 450599, 479879, 592367, 700127, 745751, 949607, 986567, 1011599, 1013399
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Mar 17 2010

Keywords

Comments

2p + 3(p + 2) = 5p + 6.
There are two parametric solutions for natural numbers:
(a) p = 5t^2 + 2t - 1, k = 5t + 1, necessarily for a prime p: t = 2s => p = 20s^2 + 4s - 1, k = 10s + 1.
If s = 3k + 2 => p of (a) is not prime but a multiple of 3.
If the least significant digit of k is 1, solution of (a) for s = (k - 1)/10).
(b) p = 5t^2 + 8t + 2, k = 5t + 4, necessarily for a prime p: t = 2s - 1 => p = 20s^2 - 4s - 1, N = 10s-1.
If s = 3k + 1 => p of (b) is not prime but a multiple of 3.
If the least significant digit of k is 9, solution of (b) for s = (k + 1)/10).

Examples

			71 and 73 are twin primes, 2 * 71 + 3 * 73 = 19^2.
191 and 193 are twin primes, 2 * 191 + 3 * 193 = 31^2.

References

  • Leonard E. Dickson, History of the Theory of numbers, vol. 2: Diophantine Analysis, Dover Publications 2005.
  • Richard K. Guy, Unsolved Problems in Number Theory, New York, Springer-Verlag, 1994.
  • Edmund Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Band I, B. G. Teubner, Leipzig u. Berlin, 1909.

Links

Crossrefs

Cf. A001359, A061308, A069496, A119859, A172271, A172494, A173255.

Programs

  • Mathematica
    Select[Prime[Range[10^5]], PrimeQ[# + 2] && IntegerQ[Sqrt[2# + 3(# + 2)]] &] (* Alonso del Arte, Dec 05 2011 *)
Select[(Range[2251]^2 - 6)/5, And @@ PrimeQ[# + {0, 2}] &] (* Amiram Eldar, Dec 24 2019 *)
Select[Partition[Prime[Range[80000]],2,1],#[[2]]-#[[1]]==2&&IntegerQ[Sqrt[ 2#[[1]]+ 3#[[2]]]]&][[All,1]] (* Harvey P. Dale, May 12 2022 *)
  • PARI
    forstep(n=1,1e4,[10,8,10,2],if(isprime(p=n^2\5-1)&&isprime(p+2),print1(p", "))) \\ Charles R Greathouse IV, Dec 05 2011

A178337 Numbers k such that (k^3 + 2, n^3 + 4) is a twin prime pair.

Original entry on oeis.org

1, 3, 45, 63, 69, 129, 363, 495, 555, 579, 885, 993, 1053, 1185, 1719, 1839, 2055, 2175, 2199, 2409, 2595, 3039, 3063, 3303, 3399, 3555, 3615, 4245, 4443, 4449, 5073, 5373, 5535, 5703, 5949, 6015, 6075, 6693, 6795, 6849, 7023, 7119, 7155, 7509, 7779, 8535
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 25 2010

Keywords

Comments

With the exception of k = 1, all k are odd multiples of 3 with a least-significant decimal digit of 3, 5 or 9.
A178336(n) gives the values of k^3 + 2.

Examples

			1^3 + 2 = 3 = prime(2) and 3+2 = prime(3) are twin primes, so n=1 is a term.
45^3 + 2 = 91127 = prime(8811) and 91127+2 = prime(8812) are twin primes, so 45 is a term.
10893^3 + 2 = 1292535591959 = prime(48144179941) is a lower twin prime, so 10893 is a term.

Links

Crossrefs

Cf. A013159, A053703, A132282, A144953, A173255, A178336.

Programs

  • Magma
    [n: n in [0..9000] | IsPrime(n^3+2) and IsPrime(n^3+4)]; // Vincenzo Librandi, Nov 18 2010
  • Mathematica
    seqQ[n_] := And @@ PrimeQ[n^3 + 3 + {-1, 1}]; Select[Range[8535], seqQ] (* Amiram Eldar, Jan 11 2020*)

Extensions

Keyword:base removed by R. J. Mathar, Jun 27 2010

A178506 Lesser of a "near cube" twin prime pair (k^3 - 4, k^3 - 2).

Original entry on oeis.org

3371, 8120597, 69426527, 108531329, 176558477, 1207949621, 2379270371, 3477265871, 3560550179, 4227952109, 8012005997, 12665630687, 13060888871, 15832158827, 15945922409, 18337088849, 20279414579, 22354272509, 30283802609, 60559558979, 70496180087, 98035951127
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 29 2010

Keywords

Comments

p + 2 = k^3 - 2 is form of "near(est) cube" prime smaller than cube number k^3, as k^3 - 1 = (k-1) * (k^2 + k + 1), only prime for k=2.

Examples

			p = 3371 = prime(475) = 15^3 - 4, (p, p+2) is twin prime pair tp(90), 3371 is the first term.

Links

Crossrefs

Cf. A178336.
Cf. A001359, A053703, A132282, A144953, A173255, A178228, A038600.

Programs

  • Mathematica
    Select[Range[10^4]^3 - 4, And @@ PrimeQ[# + {0, 2}] &] (* Amiram Eldar, Dec 25 2019 *)

Extensions

a(13) corrected and more terms from Amiram Eldar, Dec 25 2019

A173560 Numbers m such that (6*m)^5 is a sum of a twin prime pair.

Original entry on oeis.org

16, 44, 84, 135, 161, 631, 849, 880, 1035, 1086, 1721, 1815, 2155, 2704, 2871, 2975, 3011, 3159, 3220, 3365, 3390, 3669, 3996, 4075, 4704, 4761, 5025, 5090, 5299, 5585, 5640, 5970, 6314, 6606, 7035, 7785, 8104, 8129, 8610, 9116, 9665, 9966, 10249
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Feb 21 2010

Keywords

Comments

The twin prime pairs are characterized in A173255.
No such m has least significant digit (LSD) e = 2 or 7 because a = (6 * e)^5/2 - 1, representing the smaller of the twin primes, would get LSD 5.
No such m has LSD e = 3 or 8, because a+2 = (6 * e)^5/2 + 1, representing the larger prime, would get LSD 5.
The primes in this sequence here are a(6) = 631 = prime(115), a(11) = 1721 = prime(268),
a(17) = 3011 = prime(432), a(49) = 10859 = prime(1320), ...

Examples

			p = (6 * 16)^5/2 - 1 = 4076863487 = A000040(193435931); p+2 = A000040(193435932), so a(1) = 16.
p = (6 * 44)^5/2 - 1 = 641194278911 = A000040(24524572848); p+2 = A000040(24524572849), so a(2) = 44.
p = (6 * 84)^5/2 - 1 = 16260080320511 = A000040(553382827197); p+2 = A000040(553382827198), so a(3) = 84.

Links

Crossrefs

Cf. A001359, A061308, A069496, A172271, A172494, A173255.

Programs

  • Mathematica
    Select[Range[700],AllTrue[((6*#)^5-2)/2+{0,2},PrimeQ]&] (* Harvey P. Dale, Dec 21 2024 *)
  • PARI
    isok(m) = {my(k = (6*m)^5/2); isprime(k-1) && isprime(k+1);} \\ Amiram Eldar, Jul 19 2025

A178033 Lesser of a twin prime pair (p,p+2) such that permuting the digits of p and those of p+2 gives a different twin prime pair (q, q+2).

Original entry on oeis.org

281, 461, 641, 821, 1031, 1091, 1229, 1277, 1301, 1319, 1427, 1697, 1721, 1787, 1877, 2081, 2129, 2381, 2687, 2711, 2801, 3119, 3251, 3257, 3371, 3467, 3527, 3581, 3821, 3851, 4091, 4127, 4157, 4217, 4241, 4271, 4421, 4517, 4637, 4649, 4721, 4787, 4931, 4967, 5231, 5417, 5477, 5651
Offset: 1

Views

Author

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 17 2010

Keywords

Comments

Permutations with initial zeros are disallowed, so that 101 is not a member (101,103 and 11,13); equivalently, we require that p is a permutation of the digits of q as well.

Examples

			281 is a term as 281 is the lesser of the twin prime pair 281,283, and after permuting 821, 823 is also a twin prime pair.
1229 is a term as (1229,1231) is a twin prime pair and after permuting (2129, 2131) is also a twin prime pair.

Links

Crossrefs

Cf. A001359, A172271, A172494, A173255, A174454

Programs

  • Mathematica
    perm@n_ :=
 Select[FromDigits@# & /@
   DeleteCases[Rest@Permutations@IntegerDigits@n, _?(First@# == 0 &)],
   PrimeQ];
Cases[{#, perm@# & /@ #} & /@
  Cases[6*# + {-1, 1} & /@
    Range@2000, {?PrimeQ ..}], {{x, }, {{__, a_, _}, {_, b_, _}} /; b - a == 2} :> x] (* Hans Rudolf Widmer, Oct 04 2024 *)

Extensions

Corrected and edited by D. S. McNeil, Nov 23 2010
More terms from Hans Rudolf Widmer, Oct 04 2024

A283698 Numbers k such that {k^2 + 2, k^2 + 4} and {k^3 + 2, k^3 + 4} are twin prime pairs.

Original entry on oeis.org

1, 3, 45, 2055, 39033, 48585, 101535, 104553, 112383, 117723, 129315, 152553, 170793, 178095, 234483, 246435, 258093, 272403, 304845, 306885, 365343, 372663, 375813, 405393, 405975, 436425, 456903, 494193, 538965, 551475, 559713, 569805, 570033, 767895, 792903
Offset: 1

Views

Author

K. D. Bajpai, Mar 14 2017

Keywords

Comments

Except a(1), all terms are multiples of 3.
a(n) == {3 or 15} (mod 30) for n>2.

Examples

			a(2) = 3, {3^2 + 2 = 11, 3^2 + 4 = 13 } and {3^3 + 2 = 29, 3^3 + 4 = 31} are twin prime pairs.
a(3) = 45, {45^2 + 2 = 2027, 45^2 + 4 = 2029 } and {45^3 + 2 = 91127, 45^3 + 4 = 91129} are twin prime pairs.

Links

Crossrefs

Intersection of A086381 and A178337.
Cf. A000040, A144953, A173255, A178336.

Programs

  • Mathematica
    Select[Range[1000000], PrimeQ[#^2 + 2] && PrimeQ[#^2 + 4] && PrimeQ[#^3 + 2] && PrimeQ[#^3 + 4] &]
  • PARI
    for(n=1, 100000, if(isprime(n^2+2) && isprime(n^2+4) && isprime(n^3+2) && isprime(n^3+4), print1(n, ", ")))
Showing 1-7 of 7 results.

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

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