A008328 Number of divisors of prime(n)-1.

1, 2, 3, 4, 4, 6, 5, 6, 4, 6, 8, 9, 8, 8, 4, 6, 4, 12, 8, 8, 12, 8, 4, 8, 12, 9, 8, 4, 12, 10, 12, 8, 8, 8, 6, 12, 12, 10, 4, 6, 4, 18, 8, 14, 9, 12, 16, 8, 4, 12, 8, 8, 20, 8, 9, 4, 6, 16, 12, 16, 8, 6, 12, 8, 16, 6, 16, 20, 4, 12, 12, 4, 8, 12, 16, 4, 6, 18, 15, 16, 8, 24, 8

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Also the number of irreducible factors of Phi(p,x)-1, for cyclotomic polynomial Phi(p,x) and prime p. The formula is Phi(p,x)-1 = x*Product_{n>1, n|p-1} Phi(n,x). - T. D. Noe, Oct 17 2003

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• Karl Prachar, Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form p-1 haben, Monatshefte für Mathematik, Vol. 59 (1955), pp. 91-97.
• Eric Weisstein's World of Mathematics, Cyclotomic Polynomial.

Crossrefs

Cf. A000005, A006093, A066800, A103199.

Programs

Maple

for i from 1 to 500 do if isprime(i) then print(tau(i-1)); fi; od;
A008328 := proc(n)
    numtheory[tau](ithprime(n)-1) ;
end proc: # R. J. Mathar, Oct 30 2015

Mathematica

DivisorSigma[0,#-1]&/@Prime[Range[90]] (* Harvey P. Dale, Dec 08 2011 *)

PARI

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

Formula

a(n) = A000005(A006093(n)) = A066800(prime(n)). - R. J. Mathar, Oct 01 2017

From Amiram Eldar, Apr 16 2024: (Start)

Formulas from Prachar (1955):

Sum_{prime(n) < x} a(n) = x * log(log(x)) + B*x + O(x/log(x)), where B is a constant.

There is a constant c > 0 such that for infinitely many values of n we have a(n) > exp(c * log(prime(n))/log(log(prime(n)))). (End)