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

%I A008328 #43 Apr 03 2025 08:59:36
%S A008328 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,
%T A008328 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,
%U A008328 16,8,6,12,8,16,6,16,20,4,12,12,4,8,12,16,4,6,18,15,16,8,24,8
%N A008328 Number of divisors of prime(n)-1.
%C A008328 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
%H A008328 T. D. Noe, <a href="/A008328/b008328.txt">Table of n, a(n) for n = 1..10000</a>
%H A008328 Karl Prachar, <a href="https://doi.org/10.1007/BF01302992">Über die Anzahl der Teiler einer natürlichen Zahl, welche die Form p-1 haben</a>, Monatshefte für Mathematik, Vol. 59 (1955), pp. 91-97.
%H A008328 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CyclotomicPolynomial.html">Cyclotomic Polynomial</a>.
%F A008328 a(n) = A000005(A006093(n)) = A066800(prime(n)). - _R. J. Mathar_, Oct 01 2017
%F A008328 From _Amiram Eldar_, Apr 16 2024: (Start)
%F A008328 Formulas from Prachar (1955):
%F A008328 Sum_{prime(n) < x} a(n) = x * log(log(x)) + B*x + O(x/log(x)), where B is a constant.
%F A008328 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)
%p A008328 for i from 1 to 500 do if isprime(i) then print(tau(i-1)); fi; od;
%p A008328 A008328 := proc(n)
%p A008328     numtheory[tau](ithprime(n)-1) ;
%p A008328 end proc: # _R. J. Mathar_, Oct 30 2015
%t A008328 DivisorSigma[0,#-1]&/@Prime[Range[90]] (* _Harvey P. Dale_, Dec 08 2011 *)
%o A008328 (PARI) a(n) = numdiv(prime(n)-1); \\ _Michel Marcus_, Feb 25 2021
%Y A008328 Cf. A000005, A006093, A066800, A103199.
%K A008328 nonn,easy
%O A008328 1,2
%A A008328 _N. J. A. Sloane_