A069941 Number of primes p such that n! <= p <= n! + n^2.

1, 3, 3, 3, 4, 5, 5, 4, 7, 7, 9, 6, 5, 8, 4, 9, 10, 14, 8, 16, 14, 14, 7, 6, 16, 12, 12, 15, 13, 12, 9, 12, 12, 17, 13, 6, 12, 18, 15, 13, 15, 17, 15, 23, 19, 12, 13, 19, 18, 22, 20, 19, 16, 17, 19, 19, 23, 20, 18, 19, 23, 24, 19, 15, 19, 20, 26, 18, 24, 22, 24, 25, 24, 16, 23

Offset: 1

Author

Benoit Cloitre, May 04 2002

Keywords

• easy
• nonn

Comments

Conjecture: if n>=2 there are at least 3 primes p such that n!<=p<=n!+n^2 (or stronger: for n>1, a(n) > log(n)). This is stronger than the conjecture described in A037151(n). Because if n!+k is prime, k composite, k=A*B, where A and B must have, each one, at least one prime factor>n (if not: A=q*A' q<=n then n!+k is divisible by q), hence k>n^2. Also stronger (but more restrictive) than the Schinzel conjecture: "for m large enough there's at least one prime p such that m <= p <= m + log(m)^2" since n^2 < log(n!)^2 for n>5.

For the n-th term we have a(n) = pi(n!+n^2) - pi(n!), where pi(x) is the prime counting function. However, pi(n!) is difficult to compute for n>25. The Prime Number Theorem states that pi(x) and Li(x), the logarithmic integral, are asymptotically equal. Hence we can approximate a(n) by Li(n!+n^2) - Li(n!). These approximate values of a(n) are plotted as the red curve in the "Theoretical versus Actual" plot. By the way, using x/log(x) as approximation for Li(x) would change the curve by at most 1 unit. - T. D. Noe, Mar 06 2010

Links

• T. D. Noe, Table of n, a(n) for n=1..300
• T. D. Noe, Theoretical versus Actual

Crossrefs

Cf. A037151, A037153, A014085.

Programs

Mathematica

Table[Length[Select[Range[n!,n!+n^2], PrimeQ]], {n,100}] (* T. D. Noe, Mar 06 2010 *)

PARI

for(n=1,75,print1(sum(k=n!,n!+n^2,isprime(k)),","))