OFFSET
1,2
COMMENTS
Prime gaps appear to grow more slowly than any power function.
Cramér's conjecture states that prime gaps grow as follows: prime(n+1) - prime(n) = O(log(prime(n))^2).
Since prime(n) ~ n*log(n), we conjecture that a(n) > 0 for n > 0, and that the exponent 2 cannot be replaced by any smaller exponent.
Note: n*(log_2(n))^2 < n^(log(127)/log(16)) when n >= 267. Therefore the conjecture immediately above is stronger than the conjecture that A143935(n) > 0 when n > 0, which in turn is stronger than Legendre's conjecture.
This sequence relies on intervals that are slightly more than twice as wide as those in the similar sequence A166363. A comment at that sequence by Greathouse discovers zero values (representing prime-free intervals). In contrast, the present sequence 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
LINKS
Hal M. Switkay, Table of n, a(n) for n = 1..2772
Wikipedia, Cramér's conjecture
EXAMPLE
a(10) is the number of primes > 110.35 and <= 131.64. a(10) = 3, because the primes in this interval are 113, 127, and 131.
MATHEMATICA
Differences @ Table[PrimePi[n*Log2[n]^2], {n, 1, 100}] (* Amiram Eldar, Jun 27 2021 *)
PROG
(PARI) f(n) = n*(log(n)/log(2))^2;
a(n) = primepi(f(n+1)) - primepi(f(n)); \\ Michel Marcus, Jun 30 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Hal M. Switkay, Jun 27 2021
STATUS
approved