A182134 -id:A182134 - cp's OEIS frontend

A262061 Least prime(i) such that prime(i)^(1+1/i) - prime(i) > n.

2, 3, 5, 7, 11, 11, 17, 17, 23, 29, 29, 37, 41, 53, 59, 67, 79, 89, 97, 127, 127, 137, 163, 179, 211, 223, 251, 293, 307, 337, 373, 419, 479, 521, 541, 587, 691, 727, 797, 853, 929, 1009, 1151, 1201, 1277, 1399, 1523, 1693, 1777, 1931, 2053, 2203, 2333, 2521, 2647, 2953, 3119, 3299, 3527, 3847, 4127

Offset: 1

Farideh Firoozbakht and Robert G. Wilson v, Sep 11 2015

nonn

Where A246778(i) first exceeds n, stated by p_i.
Similar to A245396.
Number of terms < 10^n: 4, 19, 41, 75, 120, 176, 242, 319, 407, 506, ..., .
Concerning Firoozbakht's Conjecture (1982): (prime(n+1))^(1/(n+1)) < prime(n)^(1/n), for all n = 1 or prime(n+1) < prime(n)^(1+1/n), which can be rewritten as: (log(prime(n+1))/log(prime(n)))^n < (1+1/n)^n. This suggests a weaker conjecture: (log(prime(n+1))/log(prime(n)))^n < e.
Prime index of a(n): 1, 1, 3, 4, 5, 5, 7, 7, 9, 10, 10, 12, 13, 16, 17, 19, 22, 24, 25, 31, 31, ..., .
All terms are unique for n > 21. Indices not unique: 1 & 2, 5 & 6, 7 & 8, 10 & 11 and 20 & 21.
The distribution of initial digits, 1...9, for a(n), n<508: 140, 91, 60, 50, 44, 36, 32, 27 and 26.

			a(20) = 127 since for all primes less than the 31st prime, 127, p_k^(32/31) - p_k are less than 20.
a(100) = 38113,
a(200) = 2400407,
a(300) = 57189007,
a(400) = 828882731,
a(500) = 8748565643,
a(1000) = 91215796479037,
a(1064) = 246842748060263, limit of Mathematica by direct computation, i.e., the first Mathematica line.

Paulo Ribenboim, The little book Of bigger primes, second edition, Springer, 2004, p. 185.

Farideh Firoozbakht and Robert G. Wilson v, Table of n, a(n) for n = 1..507
Alexei Kourbatov, Upper Bounds for Prime Gaps Related to Firoozbakht's Conjecture, arXiv:1506.03042 [math.NT], 2015.
Alexei Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015

Cf. A144104, A182134, A182514, A245396, A246777, A246778, A246781, A246810, A248855, A249566.

f[n_] := Block[{p = 2, k = 1}, While[n > p^(1 + 1/k) - p, p = NextPrime@ p; k++]; p]; Array[f, 60] (* or  quicker *)
(* or quicker *) p = 2; i = 1; lst = {}; Do[ While[ p^(1 + 1/i) < n + p, p = NextPrime@ p; i++]; AppendTo[lst, p]; Print[{n, p}], {n, 100}]; lst

a(n) = {i = 0; forprime(p=2,, i++; if (p^(1+1/i) - p > n, return (p)););} \\ Michel Marcus, Oct 04 2015

Log(y) ~= g + x^(1/2) where g = Euler's Gamma.

a(2) corrected in b-file by Andrew Howroyd, Feb 22 2018

A349366 Number of primes p such that prime(n) < p <= prime(n) + (log(prime(n)))^2 - log(prime(n)) - 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 2, 2, 1, 1, 2, 2, 2, 3, 3, 2, 2, 2, 3, 4, 4, 3, 2, 1, 1, 3, 2, 3, 3, 4, 3, 3, 4, 3, 4, 5, 4, 4, 3, 2, 1, 4, 5, 4, 4, 3, 3, 3, 4, 5, 5, 5, 4, 3, 2, 2, 4, 4, 4, 3, 2, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 3, 4, 3, 4, 5, 5, 5, 5, 6, 5

Offset: 1

Hal M. Switkay, Nov 15 2021

nonn

This sequence is an example of the search for an elementary upper bound for prime gaps that is valid for all but finitely many cases. A182134 is motivated by Firoozbakht's conjecture. Kourbatov's paper proves that Firoozbakht's conjecture is equivalent to an upper bound on prime gaps of the form (log(p))^2 - log(p) - b, where 1 <= b <= 1.17. This sequence results from the choice b = 1. While Kourbatov's bound with b = 1 implies Firoozbakht's conjecture, the terms of this sequence appear to be smaller than A182134.

Conjectures: prime gaps are o((log(p))^2), but are larger infinitely often than (log(p))^(2 - epsilon), for any epsilon > 0.

			a(12) is the number of primes above prime(12), which is 37, in a gap whose width is (log(37))^2 + log(37) - 1 = 8.4278: that is, the number of primes between 37 and 45.4278, and that is 2 (namely, 41 and 43).

Hal M. Switkay, Table of n, a(n) for n = 1..4210
Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2.

Cf. A182134.

Table[Length@Select[Range[Prime@n+1,Prime@n+(Log[Prime@n])^2-Log[Prime@n]-1],PrimeQ],{n,86}] (* Giorgos Kalogeropoulos, Nov 15 2021 *)

A359930 The number of practical numbers strictly between A005153(n) and A005153(n)^(1+1/n).

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 2, 1, 3, 3, 2, 2, 2, 2, 1, 3, 4, 3, 3, 3, 2, 3, 4, 3, 3, 3, 3, 4, 3, 3, 2, 3, 3, 4, 3, 3, 3, 4, 4, 4, 4, 3, 2, 3, 4, 6, 6, 6, 5, 5, 5, 4, 4, 4, 3, 2, 3, 3, 6, 6, 5, 5, 5, 4, 4, 4, 6, 5, 6, 5, 5, 4, 4, 5, 5, 5, 5, 5, 4, 4, 4, 5, 5, 5, 6

Offset: 1

Hal M. Switkay, Jan 18 2023

nonn

This sequence is based on Sun's conjecture that A005153(n)^(1/n) is a strictly decreasing sequence for n >= 3. This conjecture states for practical numbers what Firoozbakht's conjecture says for prime numbers, and implies that a(n) > 0 for n >= 3. It is valid at least for n <= 9991. The corresponding sequence for primes is A182134.

			The tenth practical number, A005153(10), is 24. 24^(1+1/10) is approximately 32.98. There are 3 practical numbers between 24 and 32.98, namely 28, 30, and 32. Thus a(10) = 3.

Hal M. Switkay, Table of n, a(n) for n = 1..9991

Cf. A005153, A182134.

A246784 Numbers n such that both n and n+1 are in the sequence A246782.

Original entry on oeis.org

5, 6, 9, 10, 14, 22, 28, 29, 45, 216, 714573709895

Offset: 1

Farideh Firoozbakht, Oct 23 2014

Numbers n such that A182134(n) = A182134(n+1) = 2.
n = 1475067052906945 is a large term in this sequence.
a(12) > 10^12. - Robert Price, Nov 15 2014

Cf. A000040, A182134, A246782.

a(11) from Robert Price, Nov 15 2014

A246796 a(n) = A246795(n) - A246794(n) + 1.

Original entry on oeis.org

1, 1, 2, 2, 4, 3, 4, 5, 7, 4, 6, 6, 7, 9, 6, 10, 12, 7, 8, 11, 6, 13, 10, 10, 10, 10, 15, 12, 15, 15, 11, 20, 13, 13, 12, 19, 15, 22, 16, 14, 18, 17, 18, 21, 21, 21, 21, 19, 24, 22, 25, 28, 24, 27, 27, 30, 35, 30, 23, 29, 23, 28, 29, 25, 38, 24, 24, 27, 28, 27, 37, 32, 31, 39

Offset: 2

Farideh Firoozbakht, Oct 29 2014

For a(n) consecutive numbers A005669(n) - k, A246794(n) <= k <= A246795(n), A182134(A005669(n) - k) = k.

Cf. A005669, A182134, A246794, A246795.

cp's OEIS Frontend

A262061 Least prime(i) such that prime(i)^(1+1/i) - prime(i) > n.

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A349366 Number of primes p such that prime(n) < p <= prime(n) + (log(prime(n)))^2 - log(prime(n)) - 1.

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A359930 The number of practical numbers strictly between A005153(n) and A005153(n)^(1+1/n).

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A246784 Numbers n such that both n and n+1 are in the sequence A246782.

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A246796 a(n) = A246795(n) - A246794(n) + 1.

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