A084974 -id:A084974 - cp's OEIS frontend

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

Thomas Nordhaus, Feb 09 2003

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

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.

Cf. A078692, A002386, A084974 (records).

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 *)

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

A084976 Values of k that show the slow decrease in the larger values of the Andrica function Af(k) = sqrt(p(k+1)) - sqrt(p(k)), where p(k) denotes the k-th prime.

Original entry on oeis.org

4, 30, 217, 263, 367, 429, 462, 590, 650, 738, 3385, 3644, 4522, 4612, 5949, 14357, 31545, 40933, 49414, 104071, 118505, 149689, 157680, 165326, 325852, 415069, 491237, 566214, 597311, 733588, 1319945, 1736516, 2850174, 2857960, 3183065

Offset: 1

Harry J. Smith, Jun 16 2003

nonn

a(n) are values of k such that Af(k) > Af(m) for all m > k. This sequence relies on a heuristic calculation and there is no proof that it is correct.

			a(3)=217 because p(217)=1327, p(218)=1361 and Af(217) =sqrt(1361) - sqrt(1327) = 0.463722... is larger than any value of Af(m)for m>217.

R. K. Guy, "Unsolved Problems in Number Theory", Springer-Verlag 1994, A8, p. 21.
P. Ribenboim, "The Little Book of Big Primes", Springer-Verlag 1991, p. 143.

H. J. Smith, Table of n, a(n) for n=1..128
H. J. Smith, Andrica's Conjecture
Eric Weisstein's World of Mathematics, Andrica's conjecture.

Cf. A078693, A079098, A079296, A084974, A084975, A084977.

A084975 Primes that show the slow decrease in the larger values of the Andrica function Af(k) = sqrt(p(k+1)) - sqrt(p(k)), where p(k) denotes the k-th prime.

Original entry on oeis.org

11, 127, 1361, 1693, 2503, 2999, 3299, 4327, 4861, 5623, 31469, 34123, 43391, 44351, 58889, 156007, 370373, 492227, 604171, 1357333, 1562051, 2010881, 2127269, 2238931, 4652507, 6034393, 7230479, 8421403, 8917663, 11114087, 20831533

Offset: 1

Harry J. Smith, Jun 16 2003

nonn

a(n) are the primes p(k+1) such that Af(k) > Af(m) for all m > k. This sequence relies on a heuristic calculation and there is no proof that it is correct.

			a(3)=1361 because p(218)=1361, p(217)=1327 and Af(217) = sqrt(1361) - sqrt(1327) = 0.463722... is larger than any value of Af(m) for m>217.

R. K. Guy, "Unsolved Problems in Number Theory", Springer-Verlag 1994, A8, p. 21.
P. Ribenboim, "The Little Book of Big Primes", Springer-Verlag 1991, p. 143.

H. J. Smith, Table of n, a(n) for n=1..128
Eric Weisstein's World of Mathematics, Andrica's Conjecture
H. J. Smith, Andrica's Conjecture

Cf. A078693, A079098, A079296, A084974, A084976, A084977.

A084977 Values that show the slow decrease in the Andrica function Af(k) = sqrt(p(k+1)) - sqrt(p(k)), where p(k) denotes the k-th prime.

Original entry on oeis.org

670873, 639281, 463722, 292684, 260522, 256245, 244265, 228429, 215476, 213675, 203053, 167894, 144069, 137748, 119533, 108882, 92024, 81248, 63042, 56651, 52808, 52185, 36338, 36089, 35698, 29717, 27520, 26189, 23440, 23096, 23005

Offset: 1

Harry J. Smith, Jun 16 2003

nonn

a(n) = floor(1000000*Af(k)) with k such that Af(k) > Af(m) for all m > k. This sequence relies on a heuristic calculation and there is no proof that it is correct.

			a(3)=46372 because p(217)=1327, p(218)=1361 and Af(217) = sqrt(1361)- sqrt(1327) = 0.463722... is larger than any value of Af(m) for m>217.

R. K. Guy, "Unsolved Problems in Number Theory", Springer-Verlag 1994, A8, p. 21.
P. Ribenboim, "The Little Book of Big Primes", Springer-Verlag 1991, p. 143.

H. J. Smith, Table of n, a(n) for n = 1..128
H. J. Smith, Andrica's Conjecture
Eric Weisstein's World of Mathematics, Andrica's Conjecture.

Cf. A078693, A079098, A079296, A084974, A084975, A084976.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A084976 Values of k that show the slow decrease in the larger values of the Andrica function Af(k) = sqrt(p(k+1)) - sqrt(p(k)), where p(k) denotes the k-th prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A084975 Primes that show the slow decrease in the larger values of the Andrica function Af(k) = sqrt(p(k+1)) - sqrt(p(k)), where p(k) denotes the k-th prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A084977 Values that show the slow decrease in the Andrica function Af(k) = sqrt(p(k+1)) - sqrt(p(k)), where p(k) denotes the k-th prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs