

A306174


a(n) = (prime(n)^6  1)/504.


0



3515, 9577, 47892, 93345, 293722, 1180205, 1760920, 5090727, 9424810, 12542387, 21387332, 43976907, 83691535, 102222965, 179480917, 254167230, 300266322, 482316380, 648691217, 986073990, 1652722232, 2106190775, 2369151382, 2977639587, 3327579585, 4130856652
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OFFSET

5,1


COMMENTS

Note that 504 = 7*8*9. For odd prime we have p^2 == 1 (mod 8). By Fermat's theorem and Euler's totient theorem we have p^6 == 1 (mod 7) for p != 7 and p^6 == 1 (mod 9) for p != 3. So 504 divides p^6  1 for p != 2, 3, 7.
There are no primes in this sequence except for a term not shown here, which is a(3) = (5^6  1)/504 = 31. Furthermore, omega(a(n)) = A001221(a(n)) >= 4 for n >= 7, and is exactly 4 for n = 7, 8, 9, 10, 13, 14, 16, 17, 23, ...
The set of prime factors of this sequence include all primes. First, a(7) is divisible by 2, and a(8) is divisible by 3 and 7. And also, for any prime q != 2, 3, 7, by Dirichlet's theorem on arithmetic progressions, there exists a prime p of the form k*q + d with d^6 == 1 (mod p), 0 < d < q. Since gcd(q,504) = 1, we have p^6 == d^6 == 1 (mod 504*q), so q is divisible by (p^6  1)/504.
Note that a(3)=31 for prime 5 is also an integer.  Michel Marcus, Jul 05 2018


LINKS



EXAMPLE

a(5) = (11^6  1)/504 = 3515, a(6) = (13^6  1)/504 = 9577, a(7) = (17^6  1)/504 = 47892, ...


MATHEMATICA



PROG

(PARI) a(n)=(prime(n)^6  1)/504
(Magma) [(NthPrime(n)^6  1) div 504: n in [5..40]]; // Vincenzo Librandi, Jul 13 2018


CROSSREFS



KEYWORD

easy,nonn


AUTHOR



STATUS

approved



