A359070 - OEIS

%I #12 Feb 07 2023 15:16:41

%S 3,4,15,12,39,54,79,86,144,318,1591,144,20131,2014,1764,1308,46656,

%T 1296

%N Smallest k > 1 such that k^n - 1 is the product of n distinct primes.

%C a(19) > 60000 and a(20) = 3940.

%C a(19) > 5 * 10^5; a(21) = 132023; a(22) = 229430; a(24) = 4842. - _Daniel Suteu_, Dec 16 2022

%C Because of the algebraic factorization of x^n-1 (via cyclotomic polynomials), there is good reason to expect (on average) that prime values of n will have larger solutions than other numbers. That is, those values of n with many factors already get a head start by having many algebraic factors. - _Sean A. Irvine_, Jan 06 2023

%F a(n) >= A219019(n). - _Daniel Suteu_, Dec 16 2022

%e a(3) = 15 since 15^3 - 1 = 3374 = 2*7*241 is the product of 3 distinct primes and 15 is the smallest number with this property.

%o (PARI) isok(k, n) = my(f=factor(k^n - 1)); issquarefree(f) && (omega(f) == n);

%o a(n) = my(k=2); while (!isok(k, n), k++); k; \\ _Michel Marcus_, Dec 15 2022

%Y Cf. A001597, A005117, A045542, A219019, A281940, A359069.

%K nonn,hard,more

%O 1,1

%A _Kevin P. Thompson_, Dec 15 2022