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(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).
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, May 01 2009
EXTENSIONS
Edited by R. J. Mathar, May 08 2009
STATUS
approved