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A160097 Number of non-exponential divisors of n. 1
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 (list; graph; refs; listen; history; text; internal format)
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).

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: A227339 A030777 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

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Last modified September 15 18:22 EDT 2019. Contains 327082 sequences. (Running on oeis4.)