A359070 Smallest k > 1 such that k^n - 1 is the product of n distinct primes.

3, 4, 15, 12, 39, 54, 79, 86, 144, 318, 1591, 144, 20131, 2014, 1764, 1308, 46656, 1296

Offset: 1

Author

Kevin P. Thompson, Dec 15 2022

Keywords

• nonn
• hard
• more

Comments

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

a(19) > 5 * 10^5; a(21) = 132023; a(22) = 229430; a(24) = 4842. - Daniel Suteu, Dec 16 2022

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

Examples

			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.
		

Crossrefs

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

Programs

PARI

isok(k, n) = my(f=factor(k^n - 1)); issquarefree(f) && (omega(f) == n);
a(n) = my(k=2); while (!isok(k, n), k++); k; \\ Michel Marcus, Dec 15 2022

Formula

a(n) >= A219019(n). - Daniel Suteu, Dec 16 2022