A046117 -id:A046117 - cp's OEIS frontend

A104043 Primes p equal to the sum of two successive sexy primes + 1 such that p + 6 is also prime.

17, 41, 53, 101, 173, 353, 461, 1013, 1181, 1301, 1361, 1901, 2441, 4001, 4133, 4673, 4793, 5381, 5393, 5801, 6653, 10601, 11801, 12101, 12641, 12653, 15641, 15761, 16481, 19073, 21221, 23561, 23813, 23873, 25301, 25793, 25841, 25913, 26921

Offset: 1

Giovanni Teofilatto, Mar 31 2005

Cf. A023201, A046117.

A104228 INTERSECT A023201. [From R. J. Mathar, Nov 26 2008]

Removed 31 and extended by R. J. Mathar, Nov 26 2008

A104047 Primes p equal to the sum of two successive sexy primes - 1 such that p - 6 is also prime.

Original entry on oeis.org

19, 67, 79, 199, 547, 619, 739, 1459, 1759, 3319, 3739, 4027, 4567, 5107, 5419, 6367, 7219, 8719, 9187, 9907, 10459, 10867, 11119, 12547, 13099, 14827, 15739, 16927, 17047, 18307, 21319, 25939, 27259, 27367, 31327, 33967, 37579, 38839, 38959

Offset: 1

Giovanni Teofilatto, Mar 31 2005

Cf. A023201, A046117.

Select[2#+5&/@Select[Prime[Range[4200]],PrimeQ[#+6]&],And@@PrimeQ[ {#,#-6}]&] (* Harvey P. Dale, Feb 28 2012 *)

A104227 INTERSECT A046117. [From R. J. Mathar, Nov 26 2008]

23 and 29 removed, extended by R. J. Mathar, Nov 26 2008

A172988 Primes p such that either p-3 or p-6 is prime.

Original entry on oeis.org

5, 11, 13, 17, 19, 23, 29, 37, 43, 47, 53, 59, 67, 73, 79, 89, 103, 107, 109, 113, 137, 157, 163, 173, 179, 197, 199, 229, 233, 239, 257, 263, 269, 277, 283, 313, 317, 337, 353, 359, 373, 379, 389, 439, 449, 463, 467, 509, 547, 563, 569, 577, 593, 599, 607, 613

Offset: 1

Juri-Stepan Gerasimov, Feb 07 2010

nonn

5 is the only prime p for which p-3 is prime, since p-3 is even for any odd prime and 2 is the only even prime. - Harvey P. Dale, Apr 03 2019

Cf. A046117, A067831.

Select[Prime[Range[150]],AnyTrue[#+{-3,-6},PrimeQ]&] (* Requires Mathematica version 10 or later *) (* or *) Join[{5},Select[Prime[ Range[ 3,150]],PrimeQ[#-6]&]] (* see Comment *) (* Harvey P. Dale, Apr 03 2019 *)

a(n)=A046117(n+1).

449 inserted by R. J. Mathar, Mar 09 2010

A208123 Lengths of sequences of sexy primes in arithmetic progression with a common difference of six.

Original entry on oeis.org

5, 3, 3, 4, 4, 2, 3, 3, 2, 3, 3, 2, 2, 2, 3, 4, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 4, 4, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 3

Offset: 1

Eliot Ball, Mar 26 2012

No numbers in the sequence are greater than 5, and 5 only appears once, at the start.

			The first four sequences of sexy primes in arithmetic progression with common difference of six (they are 'chained together') are (5, 11, 17, 23, 29), (7, 13, 19), (31, 37, 43), (41, 47, 53, 59).

Cf. A023201, A046117.

A309158 The smallest prime, a(n), larger than prime(n) for which every even difference from 2 to prime(n) - 1 occurs at least once for some pair of primes from prime(n) to a(n) inclusive.

Original entry on oeis.org

5, 11, 13, 23, 31, 47, 47, 53, 67, 67, 73, 101, 101, 107, 113, 131, 139, 151, 151, 151, 173, 179, 193, 193, 227, 227, 233, 241, 241, 283, 283, 293, 293, 313, 313, 353, 353, 353, 353, 397, 397, 397, 421, 421, 421, 461, 461, 467, 467, 503, 503, 503, 521, 563, 569, 599, 599

Offset: 2

Sally Myers Moite, Jul 14 2019

nonn

The "prime differences prime" a(n) is the smallest prime greater than prime(n), n > 1, for which every even difference from 2 to prime(n)-1 occurs for some pair of primes from prime(n) to a(n) inclusive.
a(n) is at least prime(n) + (prime(n) - 1) = 2 * prime(n) - 1.
If the sequence of prime differences primes is infinite, there are infinitely many pairs of primes for each even difference. If there are only finitely many pairs of primes for some even difference, the sequence ends.
Ratios a(n)/prime(n), n = 2 to 15 are 1.67, 2.20, 1.86, 2.09, 2.38, 2.76, 2.47, 2.30, 2.31, 2.16, 1.97, 2.46, 2.35, 2.28.
Conjecture: The sequence is infinite.
Conjecture: There are finitely many values of n with a(n) = 2 * prime(n) - 1.
Conjecture: There are infinitely many values of n with a(n) = a(n-1).
Conjecture: For all n, a(n) <= 3 * prime(n). (This is true for n <= 101.)

			For n = 4, prime(4) = 7 and 7 - 1 = 6. Check differences for 7 and 11: 11 - 7 = 4. For 7, 11, and 13: 11 - 7 = 4, 13 - 7 = 6, 13 - 11 = 2, so a(4) = 13.
Also prime(6) = 13, 13 - 1 =  12. For 13, 17, 19, 23, 29 and 31, 29 - 17 = 12, 23 - 13 = 10, 31 - 23 = 8, 19 - 13 = 6, 17 - 13 = 4, 19 - 17 = 2, and a(6) = 31.

Cf. A000040, A006512, A046132, A046117.

for n from 2 to 58 do
   a := ithprime(n):
   for d from 2 by 2 to a - 1 do
      p := ithprime(n);
      while not isprime(p + d) do
           p := nextprime(p)
      od;
      if p + d > a then a := p + d fi
   od;
   print(n, a)
od: # Peter Luschny, Jul 17 2019

For [n=2,n <= 101,n++,
     Clear[d];d=0;
     Clear[a];a=Prime[n];
     While[d < Prime[n]-1,
           d=d+2;
           Clear[m];m=n;
           While[CompositeQ[d+Prime[m]],m++];
           If[d+Prime[m] > a,a=d+Prime[m]]];
     Print[{n,Prime[n],a,N[a/Prime[n]]}]
     ]

A341827 a(n) is the distance from n to its more distant neighboring prime.

Original entry on oeis.org

2, 1, 2, 1, 4, 3, 2, 3, 4, 1, 4, 3, 2, 3, 4, 1, 4, 3, 2, 3, 6, 5, 4, 3, 4, 5, 6, 1, 6, 5, 4, 3, 4, 5, 6, 3, 2, 3, 4, 1, 4, 3, 2, 3, 6, 5, 4, 3, 4, 5, 6, 5, 4, 3, 4, 5, 6, 1, 6, 5, 4, 3, 4, 5, 6, 3, 2, 3, 4, 1, 6, 5, 4, 3, 4, 5, 6, 3, 2, 3, 6, 5, 4, 3, 4, 5, 8

Offset: 3

Ya-Ping Lu, Feb 20 2021

nonn

a(n) is even if n is odd and vice versa. It seems that all records are even.
n - 1 and n + 1 are twin primes if a(n) = 1.
n - 2 and n + 2 are cousin primes for n > 3 if a(n) = 2.
n - 3 and n + 3 are sexy primes if a(n) = A051700(n) = 3.

Cf. A051700, A051698, A077800, A023200, A046132, A023201, A046117.

Array[Max[#1 - #2, #3 - #1] & @@ Prepend[NextPrime[#, {-1, 1}], #] &, 105, 3] (* Michael De Vlieger, Mar 17 2021 *)

for(n=3,88,my(d1=n-precprime(n-1),d2=nextprime(n+1)-n);print1(max(d1,d2),", ")) \\ Hugo Pfoertner, Mar 10 2021

from sympy import prevprime, nextprime
for n in range(3, 1001):
    prevp = prevprime(n); nextp = nextprime(n)
    print(max(n - prevp, nextp - n))

a(n) = max{n - prevprime(n), nextprime(n) - n}.

cp's OEIS Frontend

A104043 Primes p equal to the sum of two successive sexy primes + 1 such that p + 6 is also prime.

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A104047 Primes p equal to the sum of two successive sexy primes - 1 such that p - 6 is also prime.

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A172988 Primes p such that either p-3 or p-6 is prime.

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A208123 Lengths of sequences of sexy primes in arithmetic progression with a common difference of six.

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A309158 The smallest prime, a(n), larger than prime(n) for which every even difference from 2 to prime(n) - 1 occurs at least once for some pair of primes from prime(n) to a(n) inclusive.

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A341827 a(n) is the distance from n to its more distant neighboring prime.

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