A342064 -id:A342064 - cp's OEIS frontend

A342062 a(n) is the number of divisors of prime(n)^8 - 1.

8, 24, 48, 84, 96, 192, 192, 288, 224, 192, 576, 576, 672, 2304, 1024, 768, 768, 192, 768, 336, 672, 1024, 3072, 1344, 864, 576, 448, 1152, 1536, 512, 2112, 768, 1792, 768, 1152, 1344, 2304, 960, 896, 1536, 1728, 1152, 2560, 1280, 1728, 504, 1536, 2304, 1536

Offset: 1

Jon E. Schoenfield, Feb 27 2021

nonn

a(n) >= 384 for n > 20.

			   n  prime(n)      factorization of prime(n)^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     31     2^8 * 3   * 5   * 13 * 37 * 409 * 1129    576
  12     37     2^5 * 3^2 * 5   * 19 * 89 * 137 * 10529   576
  13     41     2^6 * 3   * 5   * 7 * 29^2 * 137 * 10313  672
  ...
  20     71     2^6 * 3^2 * 5   * 7 * 2521 * 12705841     336

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000040, A309906, A342063, A342064.

f:= n -> numtheory:-tau(ithprime(n)^8-1):
map(f, [$1..100]); # Robert Israel, Feb 28 2021

a[n_] := DivisorSigma[0, Prime[n]^8 - 1]; Array[a, 50] (* Amiram Eldar, Feb 27 2021 *)

a(n) = numdiv(prime(n)^8-1); \\ Michel Marcus, Feb 27 2021

a(n) = A000005(A000040(n)^8 - 1).

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

Jon E. Schoenfield, Feb 27 2021

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

			      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

Cf. A000005, A000040, A309906, A342062, A342064.

cp's OEIS Frontend

A342062 a(n) is the number of divisors of prime(n)^8 - 1.

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