A089637 -id:A089637 - cp's OEIS frontend

A181981 Duplicate of A089637.

3, 17, 41, 107, 71, 2267, 1091, 461, 1319, 1151, 347, 5741, 2999, 5279, 10139, 1487, 9461, 881, 659, 13007, 9041, 15359, 8627, 28751, 83717, 13397, 18539, 14627, 44771, 54011, 60257, 59669, 142157, 77711, 61559, 178931, 26261, 122867, 293261, 89069, 24419

Offset: 0

dead

A065044 Start of the first run of exactly n consecutive primes, none of which are twin primes.

Original entry on oeis.org

2, 47, 113, 79, 2273, 1097, 467, 1327, 1163, 353, 5749, 3011, 5297, 10151, 1493, 9467, 887, 673, 13033, 9049, 15373, 8641, 28759, 83737, 13411, 18553, 14633, 44777, 54037, 60271, 59693, 142169, 77719, 61583, 178939, 26267, 122887, 293269

Offset: 1

Jud McCranie, Nov 05 2001

nonn

Viggo Brun proved that there exist arbitrarily long sequences of primes which are not twin primes (Ribenboim, page 261).

			79 is the first prime in a run of four consecutive primes (79, 83, 89, 97), none of which are twin primes, and the next larger and smaller primes are each members of a twin pair, so a(4)=79.

Paulo Ribenboim, The New Book of Prime Number Records, Springer, 1996.

David A. Corneth and Amiram Eldar, Table of n, a(n) for n = 1..342 (terms 1..100 from Harry J. Smith)

Cf. A001097, A001359, A089637.

nextTwin(t)= { until (isprime(t + 2), t=nextprime(t + 3)); return(t) } { default(primelimit, 4294965247); for (n=1, 100, u=2; until((primepi(u) - primepi(t) - 2) == n, t=u; u=nextTwin(t)); if (n==1, t=-3); write("b065044.txt", n, " ", nextprime(t + 3)) ) } \\ Harry J. Smith, Oct 04 2009

a(n) = A151800(A089637(n) + 2), for n > 1. - Amiram Eldar, Jan 15 2020

Offset changed from 0 to 1 by Harry J. Smith, Oct 04 2009

A182048 Numbers n such that 16n^2-2n-1 and 16n^2+2n-1 are both primes.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 13, 19, 29, 32, 36, 62, 63, 70, 75, 78, 85, 93, 96, 102, 107, 109, 119, 123, 128, 145, 158, 164, 174, 177, 190, 192, 197, 219, 241, 247, 252, 280, 284, 299, 304, 318, 335, 340, 344, 354, 361, 374, 377, 382, 385, 387, 427, 434, 439, 440

Offset: 1

Gerasimov Sergey, Apr 08 2012

nonn

			a(1)=1 because 16*1^2-2*1-1=13 is prime and 16*1^2+2*1-1=17 is prime.

Harvey P. Dale, Table of n, a(n) for n = 1..3500

Cf. A171748, A089637.

Select[Range[500], PrimeQ[16 #^2 - 2 # - 1] && PrimeQ[16 #^2 + 2 # - 1] &] (* T. D. Noe, Apr 16 2012 *)
Select[Range[500],AllTrue[16#^2-1+{2#,-2#},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Jan 01 2018 *)

Corrected and extended by T. D. Noe, Apr 16 2012

A214450 Smallest prime p such that n primes exist between the prime triple (p, p+2, p+6) and the next prime triple.

Original entry on oeis.org

5, 857, 311, 17, 31391, 3461, 1427, 12917, 1997, 4517, 41, 20747, 107, 1871, 1487, 4637, 2081, 347, 7877, 23057, 80777, 1091, 18041, 641, 461, 5231, 21017, 881, 4967, 45821, 1607, 15731, 165311, 17027, 35591, 26261, 11777, 8537, 64151, 101111, 82757, 23741

Offset: 0

Michel Lagneau, Jul 18 2012

nonn

			a(3)= 17 because there exists 3 primes 29, 31 and 37 are between (17, 19,23) and (41,43,47).

Cf. A022004, A098412, A089637.

A214450 := proc(n)
local j, hi, lo ;
if n = 0 then
3;
else
for j from 1 do
hi := numtheory[pi]( A022004 (j+1)) ;
lo := numtheory[pi]( A098412 (j)) ;
if hi-lo = n+1 then
return A022004 (j);
end if;
end do:
end if;
end proc: # [Program from R. J. Mathar, adapted for this sequence (see A089637)].

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A181981 Duplicate of A089637.

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A065044 Start of the first run of exactly n consecutive primes, none of which are twin primes.

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A182048 Numbers n such that 16n^2-2n-1 and 16n^2+2n-1 are both primes.

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Mathematica

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A214450 Smallest prime p such that n primes exist between the prime triple (p, p+2, p+6) and the next prime triple.

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Maple