A246777 -id:A246777 - cp's OEIS frontend

A246778 a(n) = floor(prime(n)^(1+1/n)) - prime(n).

2, 2, 3, 4, 6, 6, 8, 8, 9, 11, 11, 12, 13, 13, 13, 14, 15, 15, 16, 16, 16, 17, 17, 18, 19, 19, 19, 19, 19, 19, 21, 21, 22, 21, 22, 22, 22, 23, 23, 23, 24, 23, 24, 24, 24, 24, 25, 26, 26, 26, 26, 26, 26, 27, 27, 27, 27, 27, 27, 27, 27, 28, 29, 29, 28, 28, 29, 30, 30, 30

Offset: 1

Farideh Firoozbakht, Sep 26 2014

nonn

The Firoozbakht Conjecture, "prime(n)^(1/n) is a strictly decreasing function of n" is true if and only if a(n) - A001223(n) is nonnegative for all n. The conjecture is true for all primes p where p < 4.0*10^18. (See A. Kourbatov link.)
0, 1, 5, 7, 10 & 20 are not in the sequence. It seems that these six integers are all the nonnegative integers which are not in the sequence.
From Alexei Kourbatov, Nov 27 2015: (Start)
Theorem: if prime(n+1) - prime(n) < prime(n)^(3/4), then every integer > 20 is in this sequence.
Proof: Let f(n) = prime(n)^(1+1/n) - prime(n). Then a(n) = floor(f(n)).
Define F(x) = log^2(x) - log(x) - 1. Using the upper and lower bounds for f(n) established in Theorem 5 of J. Integer Sequences Article 15.11.2; arXiv:1506.03042 we have F(prime(n))-3.83/(log prime(n)) < f(n) < F(prime(n)) for n>10^6; so f(n) is unbounded and asymptotically equal to F(prime(n)).
Therefore, for every n>10^6, jumps in f(n) are less than F'(x)*x^(3/4)+3.83/(log x) at x=prime(n), which is less than 1 as x >= prime(10^6)=15485863. Thus jumps in a(n) cannot be more than 1 when n>10^6. Separately, we verify by direct computation that a(n) takes every value from 21 to 256 when 30 < n <= 10^6. This completes the proof.
(End)

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

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015.
A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, arXiv:1506.03042 [math.NT], 2015.
A. Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, Journal of Integer Sequences, 18 (2015), Article 15.11.2.
Carlos Rivera, Conjecture 30
Wikipedia, Firoozbakht's conjecture.
Wikipedia, Prime gap.

Cf. A000040, A001223, A246776, A246777, A246779, A246780, A249669.

[Floor(NthPrime(n)^(1+1/n)) - NthPrime(n): n in [1..70]]; // Vincenzo Librandi, Mar 24 2015

N:= 10^4: # to get entries corresponding to all primes <= N
Primes:= select(isprime, [2,seq(2*i+1,i=1..floor((N-1)/2))]):
seq(floor(Primes[n]^(1+1/n) - Primes[n]), n=1..nops(Primes)); # Robert Israel, Mar 23 2015

f[n_] := Block[{p = Prime@ n}, Floor[p^(1 + 1/n)] - p]; Array[f, 75]

first(m)=vector(m,i,floor(prime(i)^(1+1/i)) - prime(i)) \\ Anders Hellström, Sep 06 2015

a(n) = A249669(n) - A000040(n). - M. F. Hasler, Nov 03 2014

a(n) = (log(prime(n)))^2 - log(prime(n)) + O(1), see arXiv:1506.03042. - Alexei Kourbatov, Sep 06 2015

A246776 a(n) = floor(prime(n)^(1+1/n)) - prime(n+1).

Original entry on oeis.org

1, 0, 1, 0, 4, 2, 6, 4, 3, 9, 5, 8, 11, 9, 7, 8, 13, 9, 12, 14, 10, 13, 11, 10, 15, 17, 15, 17, 15, 5, 17, 15, 20, 11, 20, 16, 16, 19, 17, 17, 22, 13, 22, 20, 22, 12, 13, 22, 24, 22, 20, 24, 16, 21, 21, 21, 25, 21, 23, 25, 17, 14, 25, 27, 24, 14, 23, 20, 28, 26

Offset: 1

Farideh Firoozbakht, Sep 26 2014

nonn

The Firoozbakht Conjecture, "prime(n)^(1/n) is a strictly decreasing function of n" is true if and only if a(n) is nonnegative for all n, n>1.
A246777 is a hard subsequence of this sequence.
18 is not in the sequence. It seems that, 18 is the only nonnegative integer which is not in the sequence.

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

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (first 4230 terms from Hal M. Switkay)
Carlos Rivera, Conjecture 30
Alexei Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015-2023.
Alexei Kourbatov, Upper Bounds for Prime Gaps Related to Firoozbakht’s Conjecture, arXiv:1506.03042 [math.NT], 2015-2019.
Alexei Kourbatov, Upper bounds for prime gaps related to Firoozbakht's conjecture, J. Int. Seq. 18 (2015) 15.11.2
Wikipedia, Prime gap.
Wikipedia, Firoozbakht Conjecture.

Cf. A000040, A001223, A005669, A246777, A246778.

Cf. A249669.

a246776 n = a249669 n - a000040 (n + 1)
-- Reinhard Zumkeller, Nov 16 2014

Table[Floor[Prime[n]^(1+1/n)]-Prime[n+1],{n,70}]

a(n) = A249669(n) - A000040(n+1). - Reinhard Zumkeller, Nov 16 2014

A246779 Strictly increasing terms of the sequence A246776: a(1)= A246776(1) and for n>0 a(n+1) is the next term greater than a(n) after that a(n) appears in A246776 for the first time.

Original entry on oeis.org

0, 1, 4, 6, 9, 11, 13, 14, 15, 17, 20, 22, 24, 25, 27, 28, 30, 32, 33, 34, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

Farideh Firoozbakht, Sep 30 2014

nonn

I conjecture that, a(n)=n+17 for all n, n>22.

Cf. A000040, A246776, A246777, A246778, A246780.

A246780 Strictly increasing terms of the sequence A246778: a(1)= A246778(1) and for n>0 a(n+1) is next term greater than a(n) after that a(n) appears in A246778 for the first time.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Farideh Firoozbakht, Sep 29 2014

nonn

I conjecture that, a(n)=n+5 for all n, n>15.

A000040, A246776, A246777, A246778, A246779.

A246789 a(n) = A246778(A005669(n)).

Original entry on oeis.org

2, 2, 4, 9, 18, 19, 34, 39, 42, 44, 74, 82, 87, 96, 129, 149, 150, 157, 184, 184, 194, 219, 259, 265, 293, 326, 343, 343, 370, 374, 418, 422, 441, 463, 468, 509, 539, 542, 548, 573, 627, 645, 659, 670, 671, 671, 687, 693, 708, 718, 750, 753, 771, 787, 845, 884, 904, 952, 999, 1040, 1055, 1169, 1193, 1193, 1428, 1446, 1475, 1547, 1552, 1579, 1590, 1601, 1604, 1657, 1704

Offset: 1

Farideh Firoozbakht, Oct 08 2014

Conjecture: prime(A005669(n))^(1+1/A005669(n)) - prime(A005669(n)) is a strictly increasing function of n.
The truth of the conjecture would imply that "this sequence is an increasing sequence" which is another conjecture not equivalent to the first conjecture.
Note that if n is in the set {1, 19, 27, 45, 63} then a(n) = a(n+1) but there is no n, where n is less than 75 and a(n+1) < a(n).

Cf. A000040, A002386, A005669, A246776, A246777, A246778.

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

Original entry on oeis.org

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

cp's OEIS Frontend

A246778 a(n) = floor(prime(n)^(1+1/n)) - prime(n).

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A246776 a(n) = floor(prime(n)^(1+1/n)) - prime(n+1).

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A246779 Strictly increasing terms of the sequence A246776: a(1)= A246776(1) and for n>0 a(n+1) is the next term greater than a(n) after that a(n) appears in A246776 for the first time.

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A246780 Strictly increasing terms of the sequence A246778: a(1)= A246778(1) and for n>0 a(n+1) is next term greater than a(n) after that a(n) appears in A246778 for the first time.

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A246789 a(n) = A246778(A005669(n)).

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

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