A090395 Denominator of d(n)/n, where d(n) is the number of divisors of n (A000005).

1, 1, 3, 4, 5, 3, 7, 2, 3, 5, 11, 2, 13, 7, 15, 16, 17, 3, 19, 10, 21, 11, 23, 3, 25, 13, 27, 14, 29, 15, 31, 16, 33, 17, 35, 4, 37, 19, 39, 5, 41, 21, 43, 22, 15, 23, 47, 24, 49, 25, 51, 26, 53, 27, 55, 7, 57, 29, 59, 5, 61, 31, 21, 64, 65, 33, 67, 34, 69, 35, 71, 6, 73, 37, 25, 38

Offset

1,3

Comments

The first occurrence of k (if it exists) is studied in A091895.

Sequence A353011 gives indices of "late birds": n such that a(k) > a(n) for all k > n. - M. F. Hasler, Apr 15 2022

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = n/g with g = A009191(n) = gcd(A000005(n), n). This explains the "rays" in the graph, e.g., g = 1 for odd squarefree n, g = 2 for even semiprimes n = 2p > 4 and n = 4p, p > 3. - M. F. Hasler, Apr 15 2022

Example

a(6) = 3 because the number of divisors of 6 is 4 and 4 divided by 6 equals 2/3, which has 3 as its denominator.

Maple

with(numtheory): seq(denom(tau(n)/n), n=1..75) ; # Zerinvary Lajos, Jun 04 2008

Mathematica

Table[ Denominator[ DivisorSigma[0, n]/n], {n, 1, 80}] (* Robert G. Wilson v, Feb 04 2004 *)

Prog

(PARI) A090395(n) = denominator(numdiv(n)/n); \\ Antti Karttunen, Sep 25 2018
(Python)
from math import gcd
from sympy import divisor_count
def A090395(n): return n//gcd(n, divisor_count(n)) # Chai Wah Wu, Jun 20 2022

Crossrefs

Cf. A000005, A090387 (numerators), A091896 (numbers not in this sequence), A353011 (indices of terms such that all subsequent terms are larger).

Sequence in context: A298734 A137926 A218342 * A168485 A276737 A270027

Adjacent sequences: A090392 A090393 A090394 * A090396 A090397 A090398

Keyword

easy,frac,nonn

Author

Ivan_E_Mayle(AT)a_provider.com, Jan 31 2004

Extensions

More terms from Robert G. Wilson v, Feb 04 2004

Status

approved