A341663 a(n) is the number of divisors of prime(n)^3 - 1.

2, 4, 6, 12, 16, 18, 10, 16, 16, 24, 24, 48, 16, 24, 16, 24, 8, 72, 72, 16, 32, 72, 16, 16, 36, 18, 24, 32, 60, 40, 32, 16, 64, 48, 48, 72, 36, 96, 8, 12, 16, 96, 96, 84, 36, 32, 192, 24, 16, 72, 32, 32, 60, 32, 36, 48, 48, 40, 144, 64, 48, 12, 64, 32, 72, 24

Offset: 1

Author

Jon E. Schoenfield, Feb 26 2021

Keywords

• nonn

Comments

a(n) >= A309906(3) = 8 for n > 3.

Examples

			        p =                   factorization
   n  prime(n)  p^3 - 1         of p^3 - 1        a(n)
  --  --------  -------  -----------------------  ----
   1      2           7  7                          2
   2      3          26  2 * 13                     4
   3      5         124  2^2 * 31                   6
   4      7         342  2 * 3^2 * 19              12
   5     11        1330  2 * 5 * 7 * 19            16
   6     13        2196  2^2 * 3^2 * 61            18
   7     17        4912  2^4 * 307                 10
   8     19        6858  2 * 3^3 * 127             16
   9     23       12166  2 * 7 * 11 * 79           16
  10     29       24388  2^2 * 7 * 13 * 67         24
  11     31       29790  2 * 3^2 * 5 * 331         24
  12     37       50652  2^2 * 3^3 * 7 * 67        48
  13     41       68920  2^3 * 5 * 1723            16
  14     43       79506  2 * 3^2 * 7 * 631         24
  15     47      103822  2 * 23 * 37 * 61          16
  16     53      148876  2^2 * 7 * 13 * 409        24
  17     59      205378  2 * 29 * 3541              8
  18     61      226980  2^2 * 3^2 * 5 * 13 * 97   72
  19     67      300762  2 * 3^2 * 7^2 * 11 * 31   72
  20     71      357910  2 * 5 * 7 * 5113          16
		

Crossrefs

Cf. A000005, A000040, A309906, A341659.

Programs

Mathematica

a[n_] := DivisorSigma[0, Prime[n]^3 - 1]; Array[a, 50] (* Amiram Eldar, Feb 26 2021 *)

PARI

a(n) = numdiv(prime(n)^3-1); \\ Michel Marcus, Feb 26 2021

Formula

a(n) = A000005(A000040(n)^3 - 1).