A008328 - OEIS

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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

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OFFSET

1,2

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.

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)

MAPLE

for i from 1 to 500 do if isprime(i) then print(tau(i-1)); fi; od;