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.

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A342063 Primes p such that p^8 - 1 has fewer than 384 divisors.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 61, 71
Offset: 1

Views

Author

Jon E. Schoenfield, Feb 27 2021

Keywords

Comments

For all primes p > 71, p^8 - 1 has at least A309906(8)=384 divisors.

Examples

			      p =
   n  a(n)        factorization of p^8 - 1         a(n)
  --  ----  -------------------------------------  ----
   1    2         3   * 5   * 17                      8
   2    3   2^5       * 5   * 41                     24
   3    5   2^5 * 3         * 13 * 313               48
   4    7   2^6 * 3   * 5^2 * 1201                   84
   5   11   2^5 * 3   * 5   * 61 * 7321              96
   6   13   2^5 * 3   * 5   * 7 * 17 * 14281        192
   7   17   2^7 * 3^2 * 5   * 29 * 41761            192
   8   19   2^5 * 3^2 * 5   * 17 * 181 * 3833       288
   9   23   2^6 * 3   * 5   * 11 * 53 * 139921      224
  10   29   2^5 * 3   * 5   * 7 * 421 * 353641      192
  11   61   2^5 * 3   * 5   * 31 * 1861 * 6922921   192
  12   71   2^6 * 3^2 * 5   * 7 * 2521 * 12705841   336
		

Crossrefs

A342064 Primes p such that p^8 - 1 has 384 divisors.

Original entry on oeis.org

821, 997, 2819, 6619, 17827, 20947, 24917, 42709, 43411, 46141, 49261, 51691, 80077, 108803, 158981, 159539, 161341, 171659, 202667, 228611, 268573, 304477, 315803, 350971, 420781, 447683, 463459, 816709, 848227, 887989, 953773, 991811, 1056829, 1131379
Offset: 1

Views

Author

Jon E. Schoenfield, Feb 27 2021

Keywords

Comments

Conjecture: sequence is infinite.
For every term p, p^8 - 1 is of the form 2^5 * 3 * 5 * q * r * s * t, where q, r, s, and t are distinct primes > 5 (see Example section).

Examples

			   p =
n  a(n)               factorization of p^8 - 1
- ----- -----------------------------------------------------
1  821  2^5 * 3 * 5 *  41 *  137 *   337021 *    227165634841
2  997  2^5 * 3 * 5 *  83 *  499 *    99401 *    494026946041
3 2819  2^5 * 3 * 5 *  47 * 1409 *  3973381 *  31575505195561
4 6619  2^5 * 3 * 5 * 331 * 1103 * 21905581 * 959708914083961
		

Crossrefs

Programs

  • Mathematica
    Select[Prime[Range[90000]],DivisorSigma[0,#^8-1]==384&] (* Harvey P. Dale, Jul 08 2025 *)
Showing 1-2 of 2 results.