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.

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

Author

Sally Myers Moite, Jul 14 2019

Keywords

• nonn

Comments

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.)

Examples

			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.
		

Crossrefs

Cf. A000040, A006512, A046132, A046117.

Programs

Maple

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

Mathematica

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]]}]
     ]