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

Author

Jon E. Schoenfield, Feb 27 2021

Keywords

• nonn

Comments

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

Examples

			   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
		

Links

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

Crossrefs

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

Programs

Maple

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

Mathematica

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

PARI

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

Formula

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