A132282 -id:A132282 - cp's OEIS frontend

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

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

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

nonn

			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.

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 (terms 1..1000 from Harvey P. Dale)

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

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

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

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

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

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

nonn

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.

			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.

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

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

[n: n in [0..9000] | IsPrime(n^3+2) and IsPrime(n^3+4)]; // Vincenzo Librandi, Nov 18 2010

seqQ[n_] := And @@ PrimeQ[n^3 + 3 + {-1, 1}]; Select[Range[8535], seqQ] (* Amiram Eldar, Jan 11 2020*)

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

A132281 Noncomposites in A067200. Noncomposites (0, 1) and primes p such that A084380(p) = p^3 + 2 is prime.

Original entry on oeis.org

0, 1, 3, 5, 29, 71, 83, 113, 173, 263, 311, 419, 431, 491, 503, 509, 683, 701, 761, 773, 839, 911, 953, 1031, 1091, 1103, 1151, 1193, 1259, 1283, 1373, 1451, 1523, 1583, 1601, 1733, 1823, 1889, 1931, 2099, 2153, 2213, 2273, 2339, 2351, 2441, 2531, 2543

Offset: 1

Jonathan Vos Post, Aug 16 2007

The corresponding near-cube primes are A132282. Analog of near-square primes. After a(1) = 0, all values must be odd. Numbers of the form n^2+2 for n=1, 2, ... are 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, ... (A059100). These are prime for indices n = 1, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, ... (A067201), corresponding to the near-square primes 3, 11, 83, 227, 443, 1091, 1523, 2027, ... (A056899). Helfgott proves with minor conditions that: "Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is squarefree." Note that 47^3 + 2 = 103825 = 5^2 * 4153 and similarly 97^3 + 2 is divisible by 5^2, but otherwise an infinite number of p^3+2 are squarefree.

			a(1) = 0 because 0^3 + 2 = 2 is prime and 0 is noncomposite.
a(2) = 1 because 1^3 + 2 = 5 is prime and 1 is noncomposite.
a(3) = 3 because 3^3 + 2 = 29 is prime and 3 is prime.
a(4) = 5 because 5^3 + 2 = 127 is prime and 5 is prime.
a(5) = 29 because 29^3 + 2 = 24391 is prime.
45 is not in the sequence because, although 45^3 + 2 = 91127 is prime, 45 is not prime.
63 is not in the sequence because, although 63^3 + 2 = 250049 is prime, 63 is not prime.
65 is not in the sequence because, although 65^3 + 2 = 274627 is prime, 65 is not prime.
a(6) = 71 because 71^3 + 2 = 357913 is prime.
a(7) = 83 because 83^3 + 2 = 571789 is prime.
a(8) = 113 because 113^3 + 2 = 1442899 is prime.
123 is not in the sequence because, although 123^3 + 2 = 1860869 is prime, 123 is not prime.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Harald Andres Helfgott, Power-free values, repulsion between points, differing beliefs and the existence of error, arXiv:0706.1497 [math.NT], 2007.

Cf. A000040, A056899, A059100, A067200, A067201, A084380, A132282.

{p in A000040 such that A067200(p) = A084380(p) = p^3 + 2 is in A000040}.

Union of {0,1} and A048637. - R. J. Mathar, Oct 18 2007

More terms from R. J. Mathar, Oct 18 2007

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

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

nonn

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.

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

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

Cf. A178336.

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

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

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

cp's OEIS Frontend

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

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

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A132281 Noncomposites in A067200. Noncomposites (0, 1) and primes p such that A084380(p) = p^3 + 2 is prime.

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

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Mathematica

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