A341656 a(n) is the number of divisors of prime(n)^4 - 1.

4, 10, 20, 36, 40, 80, 84, 60, 96, 80, 128, 120, 144, 240, 224, 160, 80, 80, 160, 144, 288, 112, 320, 288, 192, 120, 192, 240, 320, 224, 240, 160, 192, 160, 240, 288, 480, 200, 192, 320, 240, 240, 576, 288, 360, 216, 320, 256, 160, 320, 576, 560, 336, 720, 264

Offset: 1

Author

Jon E. Schoenfield, Feb 25 2021

Keywords

• nonn

Comments

a(n) >= A309906(4) = 160 for n > 26.

Examples

			        p =                       factorization
   n  prime(n)  p^4 - 1             of p^4 - 1            a(n)
  --  -------- ---------  ------------------------------  ----
   1      2           15  3 * 5                              4
   2      3           80  2^4 * 5                           10
   3      5          624  2^4 * 3 * 13                      20
   4      7         2400  2^5 * 3 * 5^2                     36
   5     11        14640  2^4 * 3 * 5 * 61                  40
   6     13        28560  2^4 * 3 * 5 * 7 * 17              80
   7     17        83520  2^6 * 3^2 * 5 * 29                84
   8     19       130320  2^4 * 3^2 * 5 * 181               60
   9     23       279840  2^5 * 3 * 5 * 11 * 53             96
  10     29       707280  2^4 * 3 * 5 * 7 * 421             80
  11     31       923520  2^7 * 3 * 5 * 13 * 37            128
  12     37      1874160  2^4 * 3^2 * 5 * 19 * 137         120
  13     41      2825760  2^5 * 3 * 5 * 7 * 29^2           144
  14     43      3418800  2^4 * 3 * 5^2 * 7 * 11 * 37      240
  15     47      4879680  2^6 * 3 * 5 * 13 * 17 * 23       224
  16     53      7890480  2^4 * 3^3 * 5 * 13 * 281         160
  17     59     12117360  2^4 * 3 * 5 * 29 * 1741           80
  18     61     13845840  2^4 * 3 * 5 * 31 * 1861           80
  19     67     20151120  2^4 * 3 * 5 * 11 * 17 * 449      160
  20     71     25411680  2^5 * 3^2 * 5 * 7 * 2521         144
  21     73     28398240  2^5 * 3^2 * 5 * 13 * 37 * 41     288
  22     79     38950080  2^6 * 3 * 5 * 13 * 3121          112
  23     83     47458320  2^4 * 3 * 5 * 7 * 13 * 41 * 53   320
  24     89     62742240  2^5 * 3^2 * 5 * 11 * 17 * 233    288
  25     97     88529280  2^7 * 3 * 5 * 7^2 * 941          192
  26    101    104060400  2^4 * 3 * 5^2 * 17 * 5101        120
		

Crossrefs

Cf. A000005, A000040, A309906.

Programs

Mathematica

a[n_] := DivisorSigma[0, Prime[n]^4 - 1]; Array[a, 50] (* Amiram Eldar, Feb 25 2021 *)

PARI

a(n) = numdiv(prime(n)^4-1); \\ Michel Marcus, Feb 25 2021

Python

from sympy import prime, divisor_count
def A341656(n): return divisor_count(prime(n)**4-1) # Chai Wah Wu, Feb 25 2021

Formula

a(n) = A000005(A000040(n)^4 - 1).