cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Original entry on oeis.org

3, 4, 15, 12, 39, 54, 79, 86, 144, 318, 1591, 144, 20131, 2014, 1764, 1308, 46656, 1296
Offset: 1

Views

Author

Kevin P. Thompson, Dec 15 2022

Keywords

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

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

A379450 a(n) is the smallest prime p such that omega(p^n - 1) is equal to n, where omega = A001221.

Original entry on oeis.org

3, 5, 7, 11, 31, 11, 79, 47, 211, 113, 2473, 47, 45841, 389, 1123, 1061
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Dec 23 2024

Keywords

Comments

a(17) <= 2487619. - Daniel Suteu, Jan 05 2025

Examples

			a(1) = 3 because omega(3^1 - 1) = omega(2) = 1,
a(2) = 5 because omega(5^2 - 1) = omega(24) = 2,
a(3) = 7 because omega(7^3 - 1) = omega(342) = 3.
		

Crossrefs

Programs

  • Mathematica
    a[n_] := Module[{p = 2}, While[PrimeNu[p^n - 1] != n, p = NextPrime[p]]; p]; Array[a, 12] (* Amiram Eldar, Dec 25 2024 *)
  • PARI
    a(n) = my(p=2); while (omega(p^n - 1) != n, p=nextprime(p+1)); p; \\ Michel Marcus, Dec 24 2024

A368163 a(n) is the smallest number k > 1 such that bigomega(k^n - 1) = n.

Original entry on oeis.org

3, 4, 4, 10, 17, 8, 25, 5, 28, 9, 81, 13, 289, 64, 100, 41, 6561, 31, 6657, 57, 529, 1025
Offset: 1

Views

Author

Daniel Suteu, Dec 14 2023

Keywords

Comments

a(23) <= 196609; a(24) = 79; a(25) <= 28561; a(26) = 14015; a(27) = 961; a(28) = 729; a(30) = 361; a(32) = 2047.

Examples

			a(5) = 17 is the smallest number of the set {k(i)} = {17, 19, 21, 26, 27, 39, 45, ...} where k(i)^5 - 1 has exactly 5 prime factors counted with multiplicity.
		

Crossrefs

Programs

  • PARI
    a(n) = my(k=2); while (bigomega(k^n-1) != n, k++); k;
Showing 1-3 of 3 results.