A249566 -id:A249566

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A246782

Numbers n such that A182134(n)=2, i.e., there exist only two primes p with prime(n) < p < prime(n)^(1+1/n).

+10
7

5, 6, 7, 9, 10, 11, 14, 15, 22, 23, 28, 29, 30, 45, 46, 61, 66, 216, 217, 367, 3793, 1319945, 1576499, 8040877, 17567976, 44405858, 445538764, 1478061204, 3643075047, 17440041685, 190836014732, 714573709895, 714573709896

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Firoozbakht's conjecture says that for every n, there exists at least one prime p such that prime(n) < p < prime(n)^(1+1/n).

Let A(m) = {n | A182134(n) = m} where A182134(n) = #{p | p is prime and prime(n) < p < prime(n)^(1+1/n)}. This sequence gives the terms of A(2) and the sequence A246781 gives the terms of A(3).

The only known indices n for which A182134(n) = 1 are {1, 2, 3, 4, 8}. It is conjectured that this is the complete set A(1).

Conjecture: For all m, where m is greater than one, A(m) is an infinite set.

a1 = 49749629143524, a2 = 1475067052906944 and a3 = 1475067052906945 are three large terms of the sequence. It is interesting that a3 - a2 = 1.

Conjecture: The sequence is infinite.

Next term is greater than 25000000.

a(34) > 10^12. - Robert Price, Nov 01 2014

The conjecture that A(1)={1, 2, 3, 4, 8} holds through 10^12. - Robert Price, Nov 01 2014

LINKS

Table of n, a(n) for n=1..33.

A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015

Wikipedia, Firoozbakht's conjecture

EXAMPLE

5 is in the sequence since there exists only two primes p, prime(5) < p < prime(5)^(1+1/5). Note that prime(5) = 11, 11^(1+1/5)) ~ 17.77 and 11 < 13 < 17 < 17.77.

MATHEMATICA

np[n_]:=(a = Prime[n]; b = a^(1 + 1/n); Length[Select[Range[a+1, b], PrimeQ]]); Do[If[np[n] == 2, Print[n]], {n, 25000000}]

PROG

(PARI) for(n=1, 9e9, 2==primepi(prime(n)^(1+1/n))-n&&print1(n", ") \\ M. F. Hasler, Nov 03 2014

(Haskell)

a246782 n = a246782_list !! (n-1)

a246782_list = filter ((== 2) . a182134) [1..]

-- Reinhard Zumkeller, Nov 17 2014

CROSSREFS

Cf. A000040, A002386, A005669, A182134, A246781, A249566.

KEYWORD

nonn,more

AUTHOR

Farideh Firoozbakht, Oct 12 2014

EXTENSIONS

a(26)-a(27) from Robert Price, Oct 24 2014

a(28)-a(33) from Robert Price, Nov 01 2014

STATUS

approved

A246781

Numbers n such that A182134(n) = 3, i.e., there exist only three primes p with prime(n) < p < prime(n)^(1 + 1/n).

+10
6

12, 13, 16, 18, 20, 21, 27, 31, 34, 39, 44, 53, 59, 60, 65, 96, 97, 98, 99, 136, 154, 202, 214, 215, 220, 221, 280, 324, 325, 326, 365, 366, 736, 780, 2146, 2225, 3792, 5946, 5947, 5948, 6902, 6903, 18524, 22078, 23510, 23511, 23512, 31542, 31544, 33606

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Firoozbakht's conjecture states that for every n, there exists at least one prime p with prime(n) < p < prime(n)^(1+1/n).

The only known indices n for which A182134(n) = 1 are {1, 2, 3, 4, 8}.

See A246782 for indices n such that A182134(n) = 2.

This sequence lists numbers n such that A182134(n) = 3.

LINKS

Robert Price, Table of n, a(n) for n = 1..170

Carlos Rivera, Conjecture 30. The Firoozbakht Conjecture PrimePuzzles.net.

A. Kourbatov, Verification of the Firoozbakht conjecture for primes up to four quintillion, arXiv:1503.01744 [math.NT], 2015

Wikipedia, Firoozbakht's conjecture

EXAMPLE

12 is in the sequence since there exists only three primes p where, prime(12) < p < prime(12)^(1 + 1/12). Note that prime(12) = 37, 37^(1 + 1/12) ~ 49.99 and 37 < 41 < 43 < 47 < 49.99.

MAPLE

N:= 10^5: # to get all terms where prime(n)^(1+1/n) < N

Primes:= select(isprime, [2, seq(2*i+1, i=1..floor((N+1)/2))]):

filter:= proc(n) local t; t:= Primes[n]^(n+1); Primes[n+3]^n <= t and Primes[n+4]^n > t end proc:

select(filter, [$1..nops(Primes)-4]); # Robert Israel, Mar 23 2015

MATHEMATICA

np[n_] := (a = Prime[n]; b = a^(1 + 1/n); Length[Select[Range[a + 1, b], PrimeQ]]); Select[Range[10000], np[#] == 3 &]

PROG

(Haskell)

a246781 n = a246781_list !! (n-1)

a246781_list = filter ((== 3) . a182134) [1..]

-- Reinhard Zumkeller, Nov 17 2014

CROSSREFS

Cf. A000040, A182134, A246782.

Cf. A249566.

KEYWORD

nonn

AUTHOR

Farideh Firoozbakht, Oct 12 2014

EXTENSIONS

a(43)-a(50) from Robert Price, Oct 24 2014

STATUS

approved

A262061

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

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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.

REFERENCES

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

LINKS

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

FORMULA

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

EXAMPLE

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.

MATHEMATICA

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

PROG

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

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

STATUS

approved

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