A172494 -id:A172494 - cp's OEIS frontend

A173255 Smaller member p of a twin prime pair (p, p+2) such that the sum p+(p+2) is a fifth power: 2*(p+1) = k^5 for some integer k.

4076863487, 641194278911, 16260080320511, 174339220049999, 420586798122287, 388931440807883087, 1715002302605720111, 2051821692518399999, 4617724356355049999, 5873208011345484287, 58698987193722272687, 76578949263222449999, 180701862444484649999, 562030251929933709311

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), Feb 14 2010

nonn

Since k^5 = 2*(p+1) is even, k is also even.
The lesser of twin primes p (except for 3) are congruent to -1 modulo 3 (see third comment in A001359), the greater of twin primes p+2 (except for 5) are congruent to 1 modulo 3. Therefore p+1 is a multiple of 3. Since k^5 = 2*(p+1) is a multiple of 3, k is also a multiple of 3. Hence k is divisible by 2 and by 3, i.e. a multiple of 6.
The lesser of twin primes except for 3 (A001359) are congruent to 1, 7 or 9 modulo 10; this applies also to the terms of the present sequence, a subsequence of A001359.

			p = 4076863487 and p+2 form a twin prime pair, their sum 8153726976 = 96^5 is a fifth power. Hence 4076863487 is in the sequence.
p = 641194278911 and p+2 form a twin prime pair, their sum 1282388557824 = 264^5 is a fifth power. Hence 641194278911 is in the sequence.
p = 388931440807883087 and p+2 form a twin prime pair, their sum 777862881615766176 = 3786^5 is a fifth power. Hence 388931440807883087 is in the sequence.
3786 is the smallest value of k that gives a prime when divided by 6, it corresponds to a(6): 3786 = 6*631 and 631 is prime. The next value of k that gives a prime when divided by 6 is 10326 and corresponds to a(11): 10326 = 6*1721 and 1721 is prime.
If p is a term and k^5 the corresponding fifth power, then a fifth-power multiple c^5*k^5 does not necessarily correspond to a term q. The fifth power 96^5 corresponds to a(1), but q = 2^5*96^5/2-1 = 130459631615 = 5*7607*3429989 is not prime, much less is (q, q+2) a twin prime pair.
If p is a term and k^5 the corresponding fifth power, and if k^5 is the product c^5*d^5 of two fifth powers where d is even, then d^5 does not necessarily correspond to a term q. The fifth power 3786^5 = 3^5*1262^5 corresponds to a(6), but q = 1262^5/2-1 = 1600540908674415 = 3*5*577*55171*3351883 is not prime, much less is (q, q+2) a twin prime pair.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001359, A006512, A014574, A061308, A069496, A119859, A172271, A172494

/* gives triples  */ [ : k in [2..10500 by 2] | IsPrime(p) and IsPrime(p+2) where p is (k^5 div 2)-1 ];

Select[Range[2, 10^5, 2]^5/2 - 1, And@@PrimeQ[# + {0, 2}] &] (* Amiram Eldar, Dec 24 2019 *)

Edited, non-specific references and keywords base, hard removed, MAGMA program added and listed terms verified by the Associate Editors of the OEIS, Feb 26 2010

More terms from Amiram Eldar, Dec 24 2019

A178228 Numbers k such that (k^3 - 2, k^3 + 2) is a pair of cousin primes (see A178227).

Original entry on oeis.org

129, 189, 369, 435, 549, 555, 561, 819, 1245, 1491, 1719, 1779, 1839, 1875, 1935, 2175, 2289, 2415, 2451, 2595, 2709, 2769, 3141, 3441, 4401, 4611, 4851, 5655, 5775, 6075, 6099, 6795, 6969, 7125, 7239, 7365, 8109, 8139, 8325, 8361, 8385, 8535, 8685, 9591, 9765

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 23 2010

nonn

Necessarily k is an odd multiple of 3, Least significant digit of k is e = 1, 5 or 9 (3^3 - 2, 7^3 + 2 are multiples of 5).

			189 is a term since 189^3 - 2 = 6751267 = prime(460792), 189^3 + 2 = 6751271 = prime(460793).
12471 is a term since 12471^3 - 2 = 1939562763109 = prime(i), i = 71166976775, 12471^3 + 2 = 1939562763113 = prime(i+1).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023200, A046132, A090121, A164834, A172494, A174370, A176130, A176229, A178227.

Select[Range[10^4], And @@ PrimeQ[#^3 + {-2, 2}] &] (* Amiram Eldar, Dec 24 2019 *)

for(n=1,10000,my(p1=n^3-2,p2=n^3+2);if(isprime(p1)&&isprime(p2)&&ispower((p1+p2)/2,3),print1(n,", "))) \\ Hugo Pfoertner, Dec 24 2019

Edited by N. J. A. Sloane, May 23 2010

a(1) and a(21) inserted by Amiram Eldar, Dec 24 2019

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

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

nonn

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).

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

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.

Amiram Eldar, Table of n, a(n) for n = 1..10000
E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, vol. 1 and vol. 2, Leipzig, Berlin, B. G. Teubner, 1909.

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

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 *)

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

A178227 Lesser of a pair (p,p+4) of cousin primes whose arithmetic mean p+2 is a cube.

Original entry on oeis.org

2146687, 6751267, 50243407, 82312873, 165469147, 170953873, 176558479, 549353257, 1929781123, 3314613769, 5079577957, 5630252137, 6219352717, 6591796873, 7245075373, 10289109373, 11993263567, 14084823373, 14724139849, 17474794873, 19880486827, 21230922607, 30988732219

Offset: 1

Ulrich Krug (leuchtfeuer37(AT)gmx.de), May 23 2010

nonn

p = n^3 - 2, p and p+4 are "near(est) cube" primes as n^3 -/+ 1 = (n -/+ 1) * (n^2 +/- n + 1).

			p = 2146687 is a term, as p + 2 = 129^3 and both p = 129^3 - 2 and p + 4 = 129^3 + 2 are prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A023200, A046132, A172494, A174370, A176130, A178228, A176229.

Select[Range[2300]^3 - 2, And @@ PrimeQ[# + {0, 4}] &] (* Amiram Eldar, Dec 24 2019 *)

isok(p) = isprime(p) && (q=nextprime(p+1)) && (q-p==4) && ispower(p+2, 3); \\ Michel Marcus, Nov 27 2016

a(n) = A178228(n)^3 - 2. - Amiram Eldar, Dec 24 2019

Corrected by D. S. McNeil, Nov 24 2010

More terms from Amiram Eldar, Dec 24 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

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

nonn

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), ...

			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.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)

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

Select[Range[700],AllTrue[((6*#)^5-2)/2+{0,2},PrimeQ]&] (* Harvey P. Dale, Dec 21 2024 *)

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

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

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.

			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.

Hans Rudolf Widmer, Table of n, a(n) for n = 1..20000

Cf. A001359, A172271, A172494, A173255, A174454

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 *)

Corrected and edited by D. S. McNeil, Nov 23 2010

More terms from Hans Rudolf Widmer, Oct 04 2024

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A173255 Smaller member p of a twin prime pair (p, p+2) such that the sum p+(p+2) is a fifth power: 2*(p+1) = k^5 for some integer k.

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A178228 Numbers k such that (k^3 - 2, k^3 + 2) is a pair of cousin primes (see A178227).

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

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A178227 Lesser of a pair (p,p+4) of cousin primes whose arithmetic mean p+2 is a cube.

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

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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).

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