A341655 a(n) is the number of divisors of prime(n)^2 - 1.

2, 4, 8, 10, 16, 16, 18, 24, 20, 32, 28, 24, 40, 32, 24, 32, 32, 32, 32, 60, 30, 48, 32, 60, 42, 48, 40, 32, 64, 48, 54, 64, 40, 64, 48, 60, 32, 40, 40, 32, 48, 96, 64, 32, 72, 90, 64, 56, 32, 64, 60, 96, 72, 96, 40, 40, 64, 96, 32, 80, 32, 48, 96, 80, 40, 32

Offset

1,1

Comments

a(n) >= A309906(2) = 32 for n > 21.

Links

Table of n, a(n) for n=1..66.

Formula

a(n) = A000005(A000040(n)^2 - 1) = A000005(A084920(n)).

Example

        p =                factorization
   n  prime(n)  p^2 - 1      of p^2 - 1      a(n)
  --  --------  -------  ------------------  ----
   1      2         3    3                     2
   2      3         8    2^3                   4
   3      5        24    2^3 * 3               8
   4      7        48    2^4 * 3              10
   5     11       120    2^3 * 3 * 5          16
   6     13       168    2^3 * 3 * 7          16
   7     17       288    2^5 * 3^2            18
   8     19       360    2^3 * 3^2 * 5        24
   9     23       528    2^4 * 3 * 11         20
  10     29       840    2^3 * 3 * 5 * 7      32
  11     31       960    2^6 * 3 * 5          28
  12     37      1368    2^3 * 3^2 * 19       24
  13     41      1680    2^4 * 3 * 5 * 7      40
  14     43      1848    2^3 * 3 * 7 * 11     32
  15     47      2208    2^5 * 3 * 23         24
  16     53      2808    2^3 * 3^3 * 13       32
  17     59      3480    2^3 * 3 * 5 * 29     32
  18     61      3720    2^3 * 3 * 5 * 31     32
  19     67      4488    2^3 * 3 * 11 * 17    32
  20     71      5040    2^4 * 3^2 * 5 * 7    60
  21     73      5328    2^4 * 3^2 * 37       30
  22     79      6240    2^5 * 3 * 5 * 13     48
  23     83      6888    2^3 * 3 * 7 * 41     32
  24     89      7920    2^4 * 3^2 * 5 * 11   60

Mathematica

Table[DivisorSigma[0, Prime[n]^2-1], {n, 66}] (* Stefano Spezia, Feb 25 2021 *)

Prog

(PARI) a(n) = numdiv(prime(n)^2-1); \\ Michel Marcus, Feb 25 2021

Crossrefs

Cf. A000005, A000040, A084920, A309906.

Sequence in context: A102431 A076919 A336659 * A171757 A350780 A128106

Adjacent sequences: A341652 A341653 A341654 * A341656 A341657 A341658

Keyword

nonn

Author

Jon E. Schoenfield, Feb 25 2021

Status

approved