The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A217864 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A217864

Number of prime numbers between floor(n*log(n)) and (n + 1)*log(n + 1).
(history; published version)

#23 by Susanna Cuyler at Sat Apr 27 05:23:20 EDT 2019

STATUS

reviewed

approved

#22 by Joerg Arndt at Sat Apr 27 04:08:15 EDT 2019

STATUS

proposed

reviewed

#21 by Michel Marcus at Sat Apr 27 03:46:34 EDT 2019

STATUS

editing

proposed

#20 by Michel Marcus at Sat Apr 27 03:46:29 EDT 2019

NAME

Number of prime numbers between floor(n*log(n)) and (n + 1)*log(n + 1).

REFERENCES

A. Brauer and H. Zeitz, Über eine zahlentheoretische Behauptung von Legendre, Sitz. Berliner Math. Gee. 29 (1930), pp. 116-125; cited in Erdos 1935.

Legendre, Sitz. Berliner Math. Gee. 29 (1930), pp. 116-125; cited in Erdos 1935.

STATUS

approved

editing

#19 by T. D. Noe at Mon Oct 15 15:30:02 EDT 2012

STATUS

editing

approved

#18 by T. D. Noe at Mon Oct 15 15:29:32 EDT 2012

MATHEMATICA

Table[s = Floor[n*Log[n]]; PrimePi[(n+1) Log[n+1]] - PrimePi[s] + Boole[PrimeQ[s]], {n, 100}] (* T. D. Noe, Oct 15 2012 *)

STATUS

proposed

editing

#17 by Charles R Greathouse IV at Mon Oct 15 15:29:07 EDT 2012

STATUS

editing

proposed

#16 by Charles R Greathouse IV at Mon Oct 15 15:27:36 EDT 2012

COMMENTS

The first conjecture follows from Dickson's conjecture. The second conjecture follows from a theorem of Brauer & Zeitz on prime gaps. - Charles R Greathouse IV, Oct 15 2012

REFERENCES

A. Brauer and H. Zeitz, Über eine zahlentheoretische Behauptung von

Legendre, Sitz. Berliner Math. Gee. 29 (1930), pp. 116-125; cited in Erdos 1935.

LINKS

Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1935-07.pdf">On the difference of consecutive primes</a>, Quart. J. Math., Oxford Ser. 6 (1935), pp. 124-128.

PROG

(PARI) a(n)=sum(k=n*log(n)\1, (n+1)*log(n+1), isprime(k)) \\ Charles R Greathouse IV, Oct 15 2012

CROSSREFS

An alternate version of A166712.

Cf. A217564, A096509, A000905, A050504, A000720.

Cf. A096509, A000905, A050504, A000720

Cf. A166712

STATUS

proposed

editing

Discussion

Mon Oct 15

15:29

Charles R Greathouse IV: The second conjecture is true. The first conjecture is almost surely true but I can't prove it.

#15 by Jon Perry at Mon Oct 15 14:16:00 EDT 2012

STATUS

editing

proposed

#14 by Jon Perry at Mon Oct 15 14:12:59 EDT 2012

CROSSREFS

Cf. A166712

Discussion

Mon Oct 15

14:15

Jon Perry: The idea is the same, but A166712 is effecively using Ceiling rather than floor. I did at one time consider using Ceiling for the second term to maximize the gap, but I didn't use it in the end. Floor is more generous to the sequences than Ceiling - Floor[nlogn] is prime infinitely often (probably), so this makes the sequence essentially different.