A103199 -id:A103199 - cp's OEIS frontend

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

N. J. A. Sloane

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

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.

Cf. A000005, A006093, A066800, A103199.

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

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

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

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)

A372092 Numbers k where records occur for d(k)/d(k+1), where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 4, 6, 12, 30, 36, 60, 180, 240, 420, 1008, 1320, 1800, 2160, 2520, 6300, 7560, 12600, 15120, 20160, 30240, 45360, 55440, 100800, 110880, 196560, 332640, 498960, 786240, 982800, 1108800, 1580040, 1940400, 1995840, 2402400, 3880800, 4324320, 11476080, 11531520

Offset: 1

Amiram Eldar, Apr 18 2024

nonn

This sequence is infinite (Schinzel, 1954).
Is a(n) = A103199(n) - 1?
From Michael De Vlieger, Apr 19 2024: (Start)
a(12) = 1008 = 2^4 * 3^2 * 7 is the smallest term that is not a product of primorials.
a(36) = 2402400 = 2^5 * 3^1 * 5^2 * 7 * 11 * 13 is the smallest term whose exponents are not nonincreasing as prime base increases (ignoring interposing nondivisor primes). (End)

Amiram Eldar, Table of n, a(n) for n = 1..69
Michael De Vlieger, Prime power decomposition of a(n), n = 1..69.
Andrzej Schinzel, Sur une propriété du nombre de diviseurs, Publ. Math. (Debrecen), Vol. 3 (1954), pp. 261-262.

Cf. A000005, A103199, A282531.

seq[kmax_] := Module[{d1 = 1, d2, rm = 0, r, s = {}}, Do[d2 = DivisorSigma[0, k]; r = d1 / d2; If[r > rm, rm = r; AppendTo[s, k-1]]; d1 = d2, {k, 2, kmax}]; s]; seq[10^6]

lista(kmax) = {my(d1 = 1, d2, rm = 0, r); for(k = 2, kmax, d2 = numdiv(k); r = d1 / d2; if(r > rm, rm = r; print1(k-1, ", ")); d1 = d2);}

cp's OEIS Frontend

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

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