login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A228098 Number of primes p > prime(n) and such that prime(n)*p < prime(n+1)^2. 4
1, 2, 1, 3, 1, 2, 1, 1, 2, 1, 3, 2, 1, 1, 2, 2, 1, 3, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 1, 4, 1, 2, 1, 3, 1, 2, 2, 1, 2, 2, 1, 4, 1, 2, 1, 2, 4, 2, 1, 1, 2, 1, 2, 2, 2, 2, 1, 3, 2, 1, 1, 4, 2, 1, 1, 2, 1, 3, 1, 1, 1, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 3, 1, 2, 1, 1, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For n > 1, a(n)+1 is the number of composite numbers < prime(n+1)^2 and removed at the n-th step of Eratosthenes' sieve. The exception for n=1 comes from prime(1)^3 = 2^3 = 8 < prime(2)^2 = 9. This does not occur any more because prime(n)^3 > prime(n+1)^2 for all n > 1.

a(n) is related to the distribution of primes around prime(n+1). High values correspond to a large gap before prime(n+1) followed by several small gaps after prime(n+1).

a(n) >= 1 for all n, because prime(n+1) always trivially satisfies the condition. The sequence tends to alternate high and low values, and takes its minimum value 1 about half the time.

a(n) is >= and almost always equal to a'(n), defined as the number of primes between prime(n+1) (inclusive) and prime(n+1) + gap(n) (inclusive), with gap(n) = prime(n+1) - prime(n) = A001223(n).

An exception is 7, for which a(7) = 3, while the following prime is 11, thus gap(7) = 4, and there are only two primes between 11 and 11 + 4 = 15. It is probably the only one, as it is easily seen that a(n) = a'(n) if gap(n) <= sqrt(2prime(n)), which is a condition a little stronger than Andrica's Conjecture: gap(n) < 2sqrt(prime(n))+1. 7 is probably a record for the ratio gap(n)/sqrt(prime(n)), and the only prime for which it is > sqrt(2) (see A079296 for an ordering of primes according to Andrica's conjecture).

LINKS

Jean-Christophe Hervé, Table of n, a(n) for n = 1..9999

C. K. Caldwell, Gaps between primes.

Eric W. Weisstein, Andrica's Conjecture

Wikipedia, Andrica's_conjecture

Marek Wolf, A note on the Andrica conjecture

EXAMPLE

a(4)=3 because prime(4)=7, prime(5)=11, 11^2=121, and 7*11 < 7*13 < 7*17 < 121 < 7*19.

MATHEMATICA

Table[PrimePi[Prime[n + 1]^2/Prime[n]] - n, {n, 100}] (* T. D. Noe, Oct 29 2013 *)

PROG

(Sage)

P=Primes()

def a(n):

....p=P.unrank(n-1)

....p1=P.unrank(n)

....L=[q for q in [p+1..p1^2] if q in Primes() and p*q<p1^2]

....return len(L)

k=100 #change k for more terms

[a(m) for m in [1..k]] # Tom Edgar, Oct 29 2013

CROSSREFS

Cf. A000040, A001223, A083140, A079296.

Sequence in context: A078734 A028293 A092782 * A174532 A089242 A185894

Adjacent sequences:  A228095 A228096 A228097 * A228099 A228100 A228101

KEYWORD

nonn,easy

AUTHOR

Jean-Christophe Hervé, Oct 26 2013

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 26 09:52 EDT 2019. Contains 322472 sequences. (Running on oeis4.)