A228098 Number of primes p > prime(n) and such that prime(n)*p < prime(n+1)^2.

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

Offset: 1

Author

Jean-Christophe Hervé, Oct 26 2013

Keywords

• nonn
• easy

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's 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(2*prime(n)), which is a condition a little stronger than Andrica's Conjecture: gap(n) < 2*sqrt(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).

Examples

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

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, arXiv:1010.3945 [math.NT], 2010.

Crossrefs

Cf. A000040, A001223, A083140, A079296.

Programs

Mathematica

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

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*qTom Edgar, Oct 29 2013