A079296 Primes ordered by decreasing value of the function p -> sqrt(q) - sqrt(p) where q is the next prime after p.

7, 113, 23, 13, 31, 3, 1327, 19, 47, 199, 139, 89, 5, 211, 293, 53, 523, 317, 61, 181, 73, 887, 1129, 83, 37, 241, 2, 43, 283, 1669, 11, 467, 1069, 337, 509, 2477, 131, 2179, 2971, 1259, 773, 1951, 1637, 409, 3271, 421, 151, 1381, 67, 839, 619, 863, 157, 17, 661, 3137

Offset

1,1

Comments

I computed a couple of thousand primes with EXCEL and ordered them accordingly. There is a very small chance that very large prime numbers will change the order of the given terms above.

This sequence only makes sense if the sequence n -> sqrt(p_(n+1)) - sqrt(p_n) is a zero-sequence which is a hard unsolved problem. See also Andrica's conjecture.

For each consecutive prime pair p < q, the number d = sqrt(q) - sqrt(p) is unique. Place d in order from greatest to least and specify p. See Table II in Wolf. A rearrangement of the primes. - Robert G. Wilson v, Oct 18 2012

Links

Donovan Johnson, Table of n, a(n) for n = 1..10000

C. K. Caldwell, Gaps between primes.

Eric W. Weisstein, Andrica's Conjecture

Wikipedia, Andrica's conjecture

Marek Wolf, A note on the Andrica conjecture, arXiv:1010.3945 [math.NT], 2010.

Mathematica

lim = 1/5; lst = {}; p = 2; q = 3; While[p < 50000, If[ Sqrt[q] - Sqrt[p] > lim, AppendTo[lst, {p, Sqrt[q] - Sqrt[p]}]]; p = q; q = NextPrime[q]]; First@ Transpose@ Sort[lst, #1[[2]] > #2[[2]] &] (* Robert G. Wilson v, Oct 18 2012 *)

Crossrefs

Cf. A078692, A002386, A084974 (records).

Sequence in context: A293456 A099153 A270121 * A081531 A142537 A084974

Adjacent sequences: A079293 A079294 A079295 * A079297 A079298 A079299

Keyword

nonn,nice

Author

Thomas Nordhaus, Feb 09 2003

Extensions

More terms from Robert G. Wilson v, Oct 18 2012

Status

approved