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A160097
Number of non-exponential divisors of n.
3
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
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
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, May 01 2009
EXTENSIONS
Edited by R. J. Mathar, May 08 2009
STATUS
approved