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

1,6

Comments

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

Links

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

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).

Example

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).

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A030777 A353375 A056595 * A252477 A029351 A178638

Adjacent sequences: A160094 A160095 A160096 * A160098 A160099 A160100

Keyword

nonn

Author

Jaroslav Krizek, May 01 2009

Extensions

Edited by R. J. Mathar, May 08 2009

Status

approved