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%I #12 Apr 03 2023 10:36:12
%S 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,
%T 12,30,95,29,59,14,79,11,59,58,37,61,59,23,96,22,43,19,131,143,50,31,
%U 55,84,30,134,86,88,77,24,87,60,28,162,227,73,37,55,248,104,174,39,65,104,143
%N Number of regular primes between p^2 and p*q, where p and q are consecutive primes.
%C If you graph a(n) versus n, an interesting pattern with random-looking fluctuations emerges.
%C As you go farther along the n-axis, greater are the number of regular primes, on average, within each interval obtained.
%C The smallest count of 1 occurs only once at the very beginning.
%C I suspect all numbers in this sequence are > 0.
%C If one could prove that there is at least 1 regular prime within each interval, this would imply that regular primes are infinite.
%C 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).
%H C. K. Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/page.php/Regular.html">Regular prime</a>
%H D. Jao, <a href="https://planetmath.org/regularprime">Regular prime</a>, PlanetMath.org.
%F 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.
%e 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.
%o (PARI) has(p)=forstep(k=2, p-3, 2, if(numerator(bernfrac(k))%p==0, return(0))); 1
%o 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
%Y Cf. A007703, A000928.
%K nonn
%O 1,2
%A _Jaspal Singh Cheema_, Mar 12 2010
%E New name from _Charles R Greathouse IV_, Nov 02 2022
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