A160097 Number of non-exponential divisors of n.

1, 1, 1, 1, 1, 3, 1, 2, 1, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 6, 1, 3, 2, 4, 1, 7, 1, 4, 3, 3, 3, 5, 1, 3, 3, 6, 1, 7, 1, 4, 4, 3, 1, 7, 1, 4, 3, 4, 1, 6, 3, 6, 3, 3, 1, 10, 1, 3, 4, 3, 3, 7, 1, 4, 3, 7, 1, 8, 1, 3, 4, 4, 3, 7, 1, 7, 2, 3, 1, 10, 3, 3, 3, 6, 1, 10, 3, 4, 3, 3, 3, 10, 1, 4, 4, 5

Offset: 1

Author

Jaroslav Krizek, May 01 2009

Keywords

• nonn

Comments

The non-exponential divisors d|n of a number n = Product_i p(i)^e(i) are divisors d not of the form Product_i p(i)^s(i), s(i)|e(i) for all i.

Examples

			a(8) = 2 because 1 and 2^2 are non-exponential divisors of 8 = 2^3. 2^2 is a non-exponential divisor because 2^2 = 4 divides 8, but the exponent 2 = s(1) does not divide the exponent 3 = e(1).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A049419, A000040, A006881, A120944, A000961, A053810.

Cf. A001620, A327837.

Programs

Mathematica

f1[p_, e_] := e + 1; f2[p_, e_] := DivisorSigma[0, e]; a[1] = 1; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n]) - Times @@ f2 @@@ fct; Array[a, 100] (* Amiram Eldar, Oct 26 2021 *)

PARI

A049419(n) = { my(f = factor(n), m = 1); for(k=1, #f~, m *= numdiv(f[k, 2])); m; } \\ After Jovovic's formula for A049419.
A160097(n) = if(1==n,n,(numdiv(n) - A049419(n))); \\ Antti Karttunen, May 25 2017

Formula

a(n) = A000005(n) - A049419(n) for n >= 2.

a(1) = 1, a(p) = 1, a(p*q) = 3, a(p*q*...*z) = 2^k - 1, where the indices are p=primes (A000040), p*q = product of two distinct primes (A006881), and generally p*q*...*z = product of k (k > 0) distinct primes (A120944).

a(p^k) = k + 1 - A000005(k), where p are primes (A000040), p^k are prime powers A000961 (n>1), k = natural numbers (A000027).

a(p^q) = q - 1, where p and q are primes (A000040), and p^q = prime powers of primes (A053810).

Sum_{k=1..n} a(k) ~ n * (log(n) + 2*A001620 - A327837 - 1). - Amiram Eldar, Feb 03 2025

Extensions

Edited by R. J. Mathar, May 08 2009