A353011 -id:A353011 - cp's OEIS frontend

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

Ivan_E_Mayle(AT)a_provider.com, Jan 31 2004

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

			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.

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

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

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

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

A090395(n) = denominator(numdiv(n)/n); \\ Antti Karttunen, Sep 25 2018

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

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

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

A091896 Numbers n such that there exists no k for which the denominator of d(k)/k is n, where d = A000005 is the number-of-divisors function.

Original entry on oeis.org

18, 30, 72, 112, 144, 243, 252, 288, 294, 336, 360, 396, 468, 504, 576, 612, 616, 625, 684, 726, 728, 792, 810, 828, 840, 936, 952, 960, 1014, 1044, 1064, 1116, 1224, 1250, 1260, 1288, 1332, 1350, 1368, 1386, 1440, 1476, 1548, 1568, 1584, 1624, 1638, 1656

Offset: 1

Robert G. Wilson v, Feb 09 2004

nonn

The number of terms <= 10^n: 0, 3, 28, 311, 3541, for n = 1, 2, 3, 4, 5.
Sequence A353011 lists the indices n such that A090395(k) > A090395(n) for all k > n. This allows one to know whether a given number is in this sequence or not. - M. F. Hasler, Apr 15 2022
Another way to confirm a 0 is by looking at A005179(m)/m. If A005179(m)/m > n then d(k) cannot be a multiple of m. - David A. Corneth, Apr 16 2022

M. F. Hasler and David A. Corneth, Table of n, a(n) for n = 1..10000 (first 3541 terms from M. F. Hasler)

Cf. A000005 (number-of-divisors function d), A005179 (smallest number with exactly n divisors), A090395 (denominator of d(n)/n), A353011 (indices of "late birds" in A090395).

Indices of zeros in A091895 (index where n occurs first in A090395, or 0 if n is not in A090395).

a = Table[0, {2000}]; Do[m = n; b = Denominator[ DivisorSigma[0, n]/n]; If[b < 2001 && a[[b]] == 0, a[[b]] = n], {n, 1, 25000000}]; Select[ Range[2000], a[[ # ]] == 0 &]

select( {is_A091896(n)=!A091895(n)}, [1..10^4] ) \\ M. F. Hasler, Apr 04 2022

Edited by M. F. Hasler, Apr 04 2022

cp's OEIS Frontend

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

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