A079296 -id:A079296 - 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.

A228098 Number of primes p > prime(n) and such that prime(n)*p < prime(n+1)^2.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 2, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 1, 4, 1, 2, 1, 3, 1, 2, 2, 1, 2, 2, 1, 4, 1, 2, 1, 2, 4, 2, 1, 1, 2, 1, 2, 2, 2, 2, 1, 3, 2, 1, 1, 4, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 3

Offset: 1

Jean-Christophe Hervé, Oct 26 2013

For n > 1, a(n)+1 is the number of composite numbers < prime(n+1)^2 and removed at the n-th step of Eratosthenes's sieve. The exception for n=1 comes from prime(1)^3 = 2^3 = 8 < prime(2)^2 = 9. This does not occur any more because prime(n)^3 > prime(n+1)^2 for all n > 1.
a(n) is related to the distribution of primes around prime(n+1). High values correspond to a large gap before prime(n+1) followed by several small gaps after prime(n+1).
a(n) >= 1 for all n, because prime(n+1) always trivially satisfies the condition. The sequence tends to alternate high and low values, and takes its minimum value 1 about half the time.
a(n) is >= and almost always equal to a'(n), defined as the number of primes between prime(n+1) (inclusive) and prime(n+1) + gap(n) (inclusive), with gap(n) = prime(n+1) - prime(n) = A001223(n).
An exception is 7, for which a(7) = 3, while the following prime is 11, thus gap(7) = 4, and there are only two primes between 11 and 11 + 4 = 15. It is probably the only one, as it is easily seen that a(n) = a'(n) if gap(n) <= sqrt(2*prime(n)), which is a condition a little stronger than Andrica's Conjecture: gap(n) < 2*sqrt(prime(n))+1. 7 is probably a record for the ratio gap(n)/sqrt(prime(n)), and the only prime for which it is > sqrt(2) (see A079296 for an ordering of primes according to Andrica's conjecture).

			a(4)=3 because prime(4)=7, prime(5)=11, 11^2=121, and 7*11 < 7*13 < 7*17 < 121 < 7*19.

Jean-Christophe Hervé, Table of n, a(n) for n = 1..9999
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. A000040, A001223, A083140, A079296.

Table[PrimePi[Prime[n + 1]^2/Prime[n]] - n, {n, 100}] (* T. D. Noe, Oct 29 2013 *)

P = Primes()
def a(n):
    p=P.unrank(n-1)
    p1=P.unrank(n)
    L=[q for q in [p+1..p1^2] if q in Primes() and p*qTom Edgar, Oct 29 2013

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.

A218015 Number of primes p such that sqrt(q) - sqrt(p) > 1/n, where q is the prime after p.

Original entry on oeis.org

0, 6, 22, 41, 75, 132, 186, 258, 330, 416, 511, 613, 724, 860, 1001, 1163, 1372, 1563, 1751, 1965, 2179, 2412, 2685, 2945, 3258, 3581, 3885, 4194, 4525, 4857, 5246, 5644, 6024, 6402, 6767, 7229, 7695, 8177, 8666, 9156, 9674, 10185, 10740, 11283, 11824

Offset: 1

Marek Wolf and Robert G. Wilson v, Oct 18 2012

nonn

Also, the number of terms by Andrica ranking which are greater than 1/n.

			a(1) = 6 because only the primes 3, 7, 13, 23, 31 and 113 satisfy the criterion.
As an example, - sqrt(3) + sqrt(5) ~= 0.50401717 which is greater than 1/2.

T. D. Noe, Table of n, a(n) for n = 1..200
Marek Wolf, A Note on the Andrica Conjecture, arXiv:1010.3945 [math.NT], 2010.

Cf. A079296, A218012, A218014.

lst = {}; p = 2; q = 3; While[p < 10^8, If[ Sqrt[q] - Sqrt[p] > 1/50, AppendTo[lst, {p, Sqrt[q] - Sqrt[p]}]]; p = q; q = NextPrime[q]]; Table[ Length@ Select[ lst, #[[2]] > 1/n &], {n, 50}]
nn = 50; t = Table[0, {nn}]; p = 2; q = 3; While[p < 10^8, n = Floor[1/(Sqrt[q] - Sqrt[p])]; If[n <= nn, t[[n]]++]; p = q; q = NextPrime[q]]; Join[{0}, Accumulate[t]] (* T. D. Noe, Oct 18 2012 *)

A218012 Decimal expansion of -sqrt(7) + sqrt(11), Andrica's Maximum A_n.

Original entry on oeis.org

6, 7, 0, 8, 7, 3, 4, 7, 9, 2, 9, 0, 8, 0, 9, 2, 5, 8, 6, 1, 3, 3, 1, 6, 9, 8, 3, 0, 3, 1, 4, 2, 6, 2, 5, 8, 2, 1, 6, 8, 2, 9, 3, 6, 2, 5, 0, 6, 9, 0, 3, 4, 1, 6, 6, 9, 0, 3, 4, 7, 6, 8, 6, 9, 1, 5, 4, 1, 5, 8, 1, 9, 3, 7, 8, 7, 6, 0, 2, 1, 8, 9, 4, 8, 4, 5, 0, 5, 1, 2, 6, 5, 3, 7, 4, 7, 0, 4, 0, 2, 9, 1, 9, 4, 7

Offset: 0

Marek Wolf and Robert G. Wilson v, Oct 18 2012

For each consecutive prime pair, p and q with p < q, d = -sqrt(p) + sqrt(q) is unique. Place d in order from greatest to least and specify p. This is the maximum d.

			0.670873479290809258613316983031426258216829362506903416690347686915...

Titu Andreescu and Dorin Andrica, Number Theory, Structures, Examples, and Problems.
Marek Wolf, A Note on the Andrica Conjecture, arXiv:1010.3945 [math.NT] (2010).

Cf. A079296, A218014, A218015.

RealDigits[- Sqrt[7] + Sqrt[11], 10, 111][[1]]

A218014 Location of the n-th prime in its Andrica ranking.

Original entry on oeis.org

27, 6, 13, 1, 31, 4, 54, 8, 3, 100, 5, 25, 155, 28, 9, 16, 243, 19, 49, 288, 21, 62, 24, 12, 75, 422, 81, 444, 84, 2, 112, 37, 580, 11, 634, 47, 53, 150, 57, 60, 788, 20, 840, 183, 872, 10, 14, 218, 1029, 228, 80, 1074, 26, 87, 92, 99, 1237, 103, 281, 1319, 29, 15, 314, 1498, 323

Offset: 1

Marek Wolf and Robert G. Wilson v, Oct 18 2012

nonn

For each consecutive prime pair p < q, d = sqrt(q) - sqrt(p) is unique. Place d in order from greatest to least and specify p.
Last appearance by prime index: 1, 5, 7, 10, 13, 17, 20, 26, 28, 33, 35, 41, 43, 45, 49, ..., .
Last appearance of a minimum prime by Andrica ranking: 2, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, ..., .
As expected, this sequence is the lesser of the twin primes beginning with the second term, 11. See A001359.

			a(1)=27 since the first prime, 2, does not show up in the ranking until the 27th term. See A218013.
a(4)=1 since the fourth prime, 7, has the maximum A_n value, see A218012; i.e., sqrt(p_n)-sqrt(p_n+1) is at a maximum.

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

Cf. A079296, A218012, A218015.

lst = {}; p = 2; q = 3; While[p < 1600000, If[ Sqrt[q] - Sqrt[p] > 1/20, AppendTo[lst, {p, Sqrt[q] - Sqrt[p]}]]; p = q; q = NextPrime[q]]; lsu = First@ Transpose@ Sort[lst, #1[[2]] > #2[[2]] &]; Table[ Position[lsu, p, 1, 1], {p, Prime@ Range@ 65}] // Flatten

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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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A228098 Number of primes p > prime(n) and such that prime(n)*p < prime(n+1)^2.

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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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A218015 Number of primes p such that sqrt(q) - sqrt(p) > 1/n, where q is the prime after p.

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A218012 Decimal expansion of -sqrt(7) + sqrt(11), Andrica's Maximum A_n.

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A218014 Location of the n-th prime in its Andrica ranking.

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