A078693 -id:A078693 - cp's OEIS frontend

A084974 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.

7, 113, 1327, 1669, 2477, 2971, 3271, 4297, 4831, 5591, 31397, 34061, 43331, 44293, 58831, 155921, 370261, 492113, 604073, 1357201, 1561919, 2010733, 2127163, 2238823, 4652353, 6034247, 7230331, 8421251, 8917523, 11113933, 20831323

Offset: 1

Harry J. Smith, Jun 16 2003

nonn

a(n) are the primes p(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)=1327 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, A084975, A084976, A084977.

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.

A079098 Conjectured values of greatest k such that for any consecutive primes q, q', k <= q < q', sqrt(q')-sqrt(q) < 1/n.

Original entry on oeis.org

1, 113, 1327, 2971, 31397, 34061, 43331, 44293, 58831, 155921, 370261, 370261, 492113, 492113, 492113, 604073, 604073, 1357201, 1561919, 2010733, 2010733, 2010733, 2010733, 2010733, 2010733, 2010733, 2010733, 2238823, 4652353, 4652353, 4652353, 4652353

Offset: 1

Rainer Rosenthal, Feb 02 2003

nonn

Inspired by Andrica's conjecture.

Each of these terms, k, has been tested to at least 100*k. - Sean A. Irvine, Jul 29 2025

R. K. Guy, "Unsolved Problems in Number Theory", Springer-Verlag 1994, A8, p. 21

Eric Weisstein's World of Mathematics, Andrica's conjecture

Cf. A038458, A074976, A078693.

More terms from Sean A. Irvine, Jul 29 2025

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.

A079063 Least k such that sqrt(prime(n+k))-sqrt(prime(n))>1.

Original entry on oeis.org

3, 3, 2, 3, 3, 3, 3, 2, 3, 3, 4, 4, 4, 3, 4, 4, 5, 4, 5, 4, 4, 4, 4, 5, 6, 5, 4, 4, 3, 3, 5, 5, 5, 5, 6, 5, 6, 5, 6, 7, 6, 5, 5, 4, 4, 4, 7, 7, 7, 6, 6, 6, 6, 8, 7, 7, 6, 5, 6, 6, 6, 5, 6, 6, 6, 6, 7, 7, 8, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 8, 8, 9, 9, 8, 8, 7, 8, 8, 8, 7, 7, 8, 7, 6, 6, 6, 5, 6, 6, 8, 8, 9, 9, 10

Offset: 1

Benoit Cloitre, Feb 02 2003

nonn

Inspired by Andrica's conjecture. If it is true, a(n)>1 for all n.

Cf. A038458, A074976, A078693

Eric Weisstein's World of Mathematics, Andrica's conjecture

a(n)=if(n<0,0,k=1; while(abs(sqrt(prime(n+k))-sqrt(prime(n)))<1,k++); k)

Conjecture: there is a constant c>0 such that for n large enough, a(n)>c*sqrt(n) and we can take c=0.4. More precisely, there are 2 constants A and B such that A=lim sup n ->infinity a(n)/sqrt(n) exists = 0.75....; B=lim inf n ->infinity a(n)/sqrt(n) exists =0.46....

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A084974 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.

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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.

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A079098 Conjectured values of greatest k such that for any consecutive primes q, q', k <= q < q', sqrt(q')-sqrt(q) < 1/n.

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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.

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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.

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A079063 Least k such that sqrt(prime(n+k))-sqrt(prime(n))>1.

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