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.
Original entry on oeis.org
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
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.
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
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.
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
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.
A161623
Greatest k for which the Andrica-like conjectural inequalities, prime(k+1)-prime(k)-(1/n)*sqrt(prime(k)) < 0, appear to fail, based on empirical evidence.
Original entry on oeis.org
30, 429, 3644, 4612, 14357, 31545, 40933, 49414, 104071, 149689, 149689, 149689, 149689, 165326, 325852, 325852, 415069, 415069, 491237, 566214
Offset: 1
For n = 1, one needs k > 30 for the inequality to hold, and it is conjectured that it holds for all k > 30. In words, the first such inequality says that we expect to see a new prime prime(k+1) between prime(k) and prime(k)+sqrt(prime(k)) for k>30.
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Block[{nn = 1500000, p, q}, Array[Set[p[#], Prime[#]] &, nn + 1]; Array[Set[q[#], (p[# + 1] - p[#])^2] &, nn]; TakeWhile[Monitor[Table[nn - LengthWhile[Table[# q[k] < p[k], {k, nn, 1, -1}], # &] &[n^2], {n, 24}], {n, k}], # < nn/2 &]] (* Michael De Vlieger, Aug 17 2022 *)
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lista(nn) = my(N=10^7, vp=primes(N), va=vector(nn)); for (n=1, nn, my(v = v=vector(N-1, k, n^2*(vp[k+1]-vp[k])^2 < vp[k])); forstep(k=N-1, 1, -1, if (!v[k], va[n] = k; break));); va; \\ Michel Marcus, Aug 17 2022
a(2) corrected, name edited and more terms from
Michel Marcus, Aug 17 2022
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