A174220 Number of regular primes between p^2 and p*q, where p and q are consecutive primes.

1, 2, 2, 4, 3, 8, 2, 6, 8, 5, 22, 15, 7, 10, 23, 21, 6, 26, 19, 7, 34, 18, 33, 38, 27, 18, 27, 12, 30, 95, 29, 59, 14, 79, 11, 59, 58, 37, 61, 59, 23, 96, 22, 43, 19, 131, 143, 50, 31, 55, 84, 30, 134, 86, 88, 77, 24, 87, 60, 28, 162, 227, 73, 37, 55, 248, 104, 174, 39, 65, 104, 143

Offset

1,2

Comments

If you graph a(n) versus n, an interesting pattern with random-looking fluctuations emerges.

As you go farther along the n-axis, greater are the number of regular primes, on average, within each interval obtained.

The smallest count of 1 occurs only once at the very beginning.

I suspect all numbers in this sequence are > 0.

If one could prove that there is at least 1 regular prime within each interval, this would imply that regular primes are infinite.

This would be very significant since "Kummer was able to prove Fermat's Last Theorem in the case where the exponent is a regular prime, a result that prior to Wiles's recent work was the only demonstration of Fermat's Last Theorem for a large class of exponents." (see Jao link).

Links

Table of n, a(n) for n=1..72.

C. K. Caldwell, The Prime Glossary, Regular prime

D. Jao, Regular prime, PlanetMath.org.

Formula

Used the table of irregular primes by T.D. Noe in A000928 to extract a longer list of regular primes from a list of odd primes.

Example

Take any pair of consecutive primes. Say the first (2,3). Square the first term, and then take the product of the two to obtain an interval (4,6). Within this interval, there is 1 regular prime, which is 5. Hence the very first term of the sequence above is 1. Similarly, the second term, 2, refers to the two regular primes 11 and 13.

Prog

(PARI) has(p)=forstep(k=2, p-3, 2, if(numerator(bernfrac(k))%p==0, return(0))); 1
a(n, p=prime(n))=my(q=nextprime(p+1), s); forprime(r=p^2, p*q, if(has(r), s++)); s \\ Charles R Greathouse IV, Nov 02 2022

Crossrefs

Cf. A007703, A000928.

Sequence in context: A324756 A373516 A324754 * A334871 A048675 A162474

Adjacent sequences: A174217 A174218 A174219 * A174221 A174222 A174223

Keyword

nonn

Author

Jaspal Singh Cheema, Mar 12 2010

Extensions

New name from Charles R Greathouse IV, Nov 02 2022

Status

approved