A144140 -id:A144140 - cp's OEIS frontend

A166363 Number of primes in the half-open interval (n(log(n))^2..(n+1)(log(n+1))^2].

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

Offset: 1

Daniel Forgues, Oct 12 2009

nonn

The open-closed half-open intervals form a partition of the real line for x > 0, thus each prime appears in a unique interval.
The n-th interval length is ~ (log(n+1/2))^2 + 2*log(n+1/2) ~ (log(n))^2 as n goes to infinity.
The n-th interval prime density is ~ 1/[log(n+1/2)+2*log(log(n+1/2))] ~ 1/log(n) as n goes to infinity.
The expected number of primes in the n-th interval is ~ [(log(n+1/2))^2 + 2*log(n+1/2)] / [log(n+1/2)+2*log(log(n+1/2))] ~ log(n) as n goes to infinity.
For n = 1 there is no prime.
If it can be proved that each interval always contains at least one prime, this would constitute even shorter intervals than A166332(n), let alone A143898(n), as n gets large.
The Shanks Conjecture and the Cramer-Granville Conjecture tell us that the intervals of length (log(n))^2 are of very critical length (the constant M > 1 of the Cramer-Granville Conjecture definitely matters!). There seems to be some risk that one such interval does not contain a prime.
The Wolf Conjecture (which agrees better with numerical evidence) seems more in favor of each interval's containing at least one prime.
From Charles R Greathouse IV, May 13 2010: (Start)
Not all intervals > 1 contain primes!
a(n) = 0 for n = 1, 4977, 17512, 147127, 76082969 (and no others up to 10^8).
Higher values include 731197850, 2961721173, 2103052050563, 188781483769833, 1183136231564246 but this list is not exhaustive.
The intervals have length (log n)^2 + 2*log n + o(1). In the Cramer model, the probability that a given integer in the interval would be prime is approximately 1/(log n + 2*log log n). Tedious calculation gives the probability that a(n) = 0 in the Cramer model as 3C(log n)^2/n * (1 + o(1)) with C = exp(-5/2)/3. Thus under that model we would expect to find roughly C*(log N)^3 numbers n up to N with a(n) = 0. In fact, the numbers are not that common since the probabilities are not independent.
(End)
The similar sequence A345755 relies on intervals that are slightly more than twice as wide as those in the present sequence. A345755 does not include zero entries for n <= 2772, suggesting that the lengths of prime gaps may be bracketed by the two sequences. We conjecture that prime gaps may be larger than log(p)^2, but are not larger than log_2(p)^2. - Hal M. Switkay, Aug 29 2023

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime Gaps.
Eric Weisstein's World of Mathematics, Cramer-Granville Conjecture.
Eric Weisstein's World of Mathematics, Shanks Conjecture (and Wolf Conjecture.)

Cf. A166332, A000720, A111943, A143898, A134034, A143935, A144140 (primes between successive n^K, for different K), A014085 (primes between successive squares).

Cf. A182315, A345755.

a(n)=sum(k=ceil(n*log(n)^2),floor((n+1)*log(n+1)^2), isprime(k)) \\ Charles R Greathouse IV, Aug 21 2015

a(n) = pi((n+1)*(log(n+1))^2) - pi(n*(log(n))^2) since the intervals are half-open properly.

Edited by Daniel Forgues, Oct 18 2009 and Nov 01 2009

Edited by Charles R Greathouse IV, May 13 2010

A166332 Number of primes in (n^(3/2)(log(n))^(1/2)..(n+1)^(3/2)(log(n+1))^(1/2)] semi-open intervals, n >= 1.

Original entry on oeis.org

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

Offset: 1

Daniel Forgues, Oct 12 2009

nonn

Number of primes in (n*(n*log(n))^(1/2)..(n+1)*((n+1)*log(n+1))^(1/2)] semi-open intervals, n >= 1.
The semi-open intervals form a partition of the real line for x > 0, thus each prime appears in a unique interval.
a(n) = pi((n+1)^(3/2)*(log(n+1))^(1/2)) - pi(n^(3/2)*(log(n))^(1/2)) since the intervals are semi-open properly.
The n-th interval length is: ~ (1/2)*(n+1/2)^(1/2)*(3*(log(n+1/2))^(1/2)+(log(n+1/2))^(-1/2)) ~ (3/2)*n^(1/2)*(log(n))^(1/2) as n goes to infinity.
The n-th interval prime density is: ~ 2/(3*log(n+1/2)+log(log(n+1/2))) ~ 2/(3*log(n)) as n goes to infinity.
The expected number of primes for n-th interval is: ~ (n+1/2)^(1/2)*(3*(log(n+1/2))^(1/2)+(log(n+1/2))^(-1/2))/ (3*log(n+1/2)+log(log(n+1/2))) ~ n^(1/2)/(log(n))^(1/2) as n goes to infinity.
Using Excel 2003, for n in [1..1123], I obtain a(n) >= 1 (at least one prime per interval).
CAUTION: I will submit the b-file, but since Excel 2003 is limited to 15-digit precision, the rounding might assign to the wrong interval a prime which is extremely close to the limit of 2 successive intervals. The b-file NEEDS TO BE VERIFIED using interval arithmetic! (SEE NEXT)
CAUTION (ADDENDA): for n in [1..1123], the minimum ratio of... ABS(n^(3/2)*(log(n))^(1/2)-ROUND(n^(3/2)*(log(n))^(1/2)))/(n^(3/2)*(log(n))^(1/2)) that I got is 5.04999E-09 which is well above 1E-15 (15-digit limit of Excel 2003), so no interval ended too close to an integral value and every prime has then been assigned to its proper interval. My b-file should then be reliable.
If it can be proved that each interval always contains at least one prime, this would constitute shorter intervals than A143898(n) as n gets large.
The sequence A166363 gives even shorter intervals that seem to always contain at least one prime (for n > 1)!

Daniel Forgues, Table of n, a(n) for n = 1..1123

Cf. A143898, A134034, A143935 (for primes between successive n^K, for different K).
Cf. A144140 (showing that for n^K, K=3/2, some intervals fails to contain primes).
Cf. A166363 (for primes in even shorter intervals).
Cf. A014085 (for primes between successive squares).
Cf. A000720.

Corrected and edited by Daniel Forgues, Oct 14 2009

Edited by Daniel Forgues, Oct 20 2009

A144137 Numbers n such that between n^K and (n+1)^K there are no primes, where K = sqrt(2).

Original entry on oeis.org

4, 24, 29, 33, 40, 43, 56, 59, 84, 117, 122, 128, 132, 139, 145, 156, 162, 163, 183, 190, 203, 230, 253, 257, 286, 297, 303, 306, 315, 319, 336, 371, 403, 420, 433, 447, 456, 467, 479, 537, 543, 563, 592, 595, 599, 624, 699, 746, 755, 767, 774, 782, 805, 814

Offset: 1

Artur Jasinski, Sep 11 2008

nonn

			a(1)=4 because in range 4^sqrt(2) = 7.10299... and (4+1)^sqrt(2) = 9.73852... there are no primes (8 and 9 aren't primes).

T. D. Noe, Table of n, a(n) for n=1..133 (no other n < 10^6)

Cf. A014085, A143898, A143935, A144140.

a = {}; k = Sqrt[2]; Do[If[Length[Select[Range[Ceiling[n^k], Floor[(n + 1)^k]], PrimeQ]] == 0, AppendTo[a, n]], {n, 3000}]; a
Select[Range[850],PrimePi[(#+1)^Sqrt[2]]-PrimePi[#^Sqrt[2]]==0&] (* or *) SequencePosition[PrimePi[Range[850]^Sqrt[2]],{x_,x_}][[All,1]] (* Harvey P. Dale, Jul 31 2021 *)

A191858 The greatest integer M for which there are no primes between M^(1+1/n) and (M+1)^(1+1/n).

Original entry on oeis.org

0, 1051, 6776941, 50904310155, 833954771945899

Offset: 1

Alexei Kourbatov, Jun 18 2011

Terms are conjectural, even under the Riemann Hypothesis.
(1) The initial term a(1)=0 gives a simple restatement of Legendre's conjecture: There are no primes between 0^2 and 1^2, but there is a prime between m^2 and (m+1)^2 for m>0.
(2) Lists of known maximum prime gaps and known first occurrences of prime gaps help verify the initial terms in this sequence. However, a lengthy computation would be needed for subsequent terms.

			The second term is a(2)=1051 because there are no primes between 1051^(3/2) and 1052^(3/2), but there is at least one prime between m^(3/2) and (m+1)^(3/2) for m>1051.

Thomas R. Nicely, First occurrence prime gaps [For local copy see A000101]
Tomás Oliveira e Silva, Gaps between consecutive primes

Cf. A144140, A192870.

A195100 Numbers n such that there are no primes between (n-1)sqrt(n-1) and nsqrt(n).

Original entry on oeis.org

1, 11, 21, 25, 28, 33, 66, 122, 140, 142, 188, 307, 322, 349, 1007, 1052

Offset: 1

Juri-Stepan Gerasimov, Sep 09 2011

Cramér's conjecture implies that the sequence is finite. - Robert Israel, Aug 11 2014

No more terms up to 2*10^10. - Jinyuan Wang, Mar 22 2019

			a(1) = 1 because there are no numbers between (1-1)*sqrt(1-1) = 0 and 1*sqrt(1) = 1.
a(2) = 11 because (11-1)*sqrt(11-1) < (nonprimes 32,33,34,35,36) < 11*sqrt(11).

H. Cramér, On the order of magnitude of the difference between consecutive prime numbers, Acta Arith. 2 (1936), 23-46.

Cf. A001223, A144140, A194852, A195008.

Primes:= select(isprime,{2,seq(2*i+1,i=1..10^6)}):
C:= map(p -> ceil(p^(2/3)), Primes);
{$1..max(C)} minus C; # Robert Israel, Aug 10 2014

Select[Range[5000], (PrimePi[# Sqrt[#]] - PrimePi[(# - 1)Sqrt[# - 1]]) == 0 &] (* Alonso del Arte, Sep 09 2011 *)
Join[{1},Flatten[Position[Partition[Table[PrimePi[n Sqrt[n]],{n,1100}], 2,1], ?(#[[2]]-#[[1]]==0&),1,Heads->False]]+1] (* _Harvey P. Dale, May 11 2018 *)

for(n=1,2*10^6,if(#primes([(n-1)*sqrt(n-1),n*sqrt(n)])==0,print1(n,", "))) \\ Derek Orr, Aug 10 2014

isok(n) = {k=floor((n-1)*sqrt(n-1))+1;while(!isprime(k),k++);k>n*sqrt(n);} \\ Jinyuan Wang, Mar 22 2019

a(n+1) = A144140(n) + 1. - Jinyuan Wang, Mar 22 2019

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A166363 Number of primes in the half-open interval (n*(log(n))^2..(n+1)*(log(n+1))^2].

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A166332 Number of primes in (n^(3/2)*(log(n))^(1/2)..(n+1)^(3/2)*(log(n+1))^(1/2)] semi-open intervals, n >= 1.

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A144137 Numbers n such that between n^K and (n+1)^K there are no primes, where K = sqrt(2).

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A191858 The greatest integer M for which there are no primes between M^(1+1/n) and (M+1)^(1+1/n).

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A195100 Numbers n such that there are no primes between (n-1)*sqrt(n-1) and n*sqrt(n).

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A166363 Number of primes in the half-open interval (n(log(n))^2..(n+1)(log(n+1))^2].

A166332 Number of primes in (n^(3/2)(log(n))^(1/2)..(n+1)^(3/2)(log(n+1))^(1/2)] semi-open intervals, n >= 1.

A195100 Numbers n such that there are no primes between (n-1)sqrt(n-1) and nsqrt(n).