A090395 -id:A090395 - cp's OEIS frontend

A353011 Indices of "late birds" in A090395 (denominator of d(n)/n): indices n such that A090395(k) > A090395(n) for all k > n.

2, 12, 24, 36, 60, 72, 84, 96, 108, 180, 240, 252, 360, 480, 504, 720, 792, 1260, 1440, 1680, 1800, 2160, 2340, 2640, 3360, 3600, 5040, 5280, 6720, 7920, 10080, 12600, 15120, 15840, 18480, 20160, 21840, 25200, 30240, 36960, 40320, 43680, 55440, 60480, 65520

Offset: 1

M. F. Hasler, Apr 15 2022

nonn

A090395(n) is the denominator of d(n)/n, where d = A000005 is the number of divisors.
The present sequence gives the indices of those terms of A090395 such that all subsequent terms are larger. This can be used to verify whether a number N is in A091896, which lists the numbers that don't occur in A090395.
It appears that a(n) is divisible by 12 for all n >= 2, by 5 for all n >= 18, by 24 (thus by 120) for all n > 23. Can somebody prove this?

Cf. A000005 (number of divisors), A090395 (denominator of A000005(n)/n), A091895 (index of first occurrence of n in A090395), A091896 (numbers that don't occur in A090395).

L=List(); forstep(n=m=65520,1,-1, m>(m=min(A090395(n),m)) && listput(L,n));Vecrev(L)

a(n+1) > a(n).

A090387 Numerator of d(n)/n, where d(n) (A000005) is the number of divisors of n.

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 2, 1, 1, 2, 2, 1, 2, 2, 4, 5, 2, 1, 2, 3, 4, 2, 2, 1, 3, 2, 4, 3, 2, 4, 2, 3, 4, 2, 4, 1, 2, 2, 4, 1, 2, 4, 2, 3, 2, 2, 2, 5, 3, 3, 4, 3, 2, 4, 4, 1, 4, 2, 2, 1, 2, 2, 2, 7, 4, 4, 2, 3, 4, 4, 2, 1, 2, 2, 2, 3, 4, 4, 2, 1, 5, 2, 2, 1, 4, 2, 4, 1, 2, 2, 4, 3, 4, 2, 4, 1, 2, 3, 2, 9, 2, 4, 2, 1, 8

Offset: 1

Ivan_E_Mayle(AT)a_provider.com, Jan 31 2004

Values of n at which k first occurs, for k >= 1: 1, 3, 4, 15, 16, 175, 64, 105, 100, 567, 1024, 1925, 4096, 3645, 784, 945, 65536, ... - Robert G. Wilson v, Feb 04 2004. [Is this A136641? - Editors]

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

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

Cf. A000005, A090395 (denominators), A136641.

with(numtheory): seq(numer(tau(n)/n), n=1..105) ; # Zerinvary Lajos, Jun 04 2008

A090387(n) = numerator(numdiv(n)/n); \\ Antti Karttunen, Sep 25 2018

from math import gcd
from sympy import divisor_count
def A090387(n): return (d := divisor_count(n))//gcd(n,d) # Chai Wah Wu, Jun 20 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

A091895 Least number k such that the denominator of d(k)/k = n, or zero if no such number exists, where d is the number-of-divisors function A000005.

Original entry on oeis.org

1, 8, 3, 4, 5, 72, 7, 80, 108, 20, 11, 240, 13, 28, 15, 16, 17, 0, 19, 480, 21, 44, 23, 48, 25, 52, 27, 560, 29, 0, 31, 448, 33, 68, 35, 864, 37, 76, 39, 160, 41, 1680, 43, 880, 540, 92, 47, 144, 49, 200, 51, 1040, 53, 972, 55, 112, 57, 116, 59, 1920, 61, 124, 756, 64, 65

Offset: 1

Robert G. Wilson v, Feb 09 2004

nonn

k is a multiple of n.
A search limit of 2*n^2 (as suggested by Hugo Pfoertner on the SeqFan list) appears to be sufficient: Up to n = 10^5, the largest ratio r(n) = a(n)/n is r(90090) = 672. - M. F. Hasler, Apr 04 2022
It appears that even a(n) <= 16*n^(4/3), verified up to n = 10^6 with search limit 2*n^2. Large values of a(n)/n^(4/3) are reached in particular at multiples of 2*3*5*7*11, but also at 2^3*3^3*5*11*13. See A352834 for more. - M. F. Hasler, Apr 15 2022

M. F. Hasler, Table of n, a(n) for n = 1..1000

Cf. A090395, zeros are in A091896.
Cf. A000005 (number-of-divisors function).
Cf. A352834 (a(n)/n).

a = Table[0, {100}]; Do[b = Denominator[DivisorSigma[0, n]/n]; If[b < 101 && a[[b]] == 0, a[[b]] = n], {n, 1, 2640}]; a

apply( {A091895(n,L=n^2*2)=forstep(k=n,L,n,denominator(numdiv(k)/k)==n&&return(k))}, [1..99]) \\ M. F. Hasler, Apr 04 2022

a(n) = n*A352834(n). - M. F. Hasler, Apr 15 2022

A348172 a(n) is the number of positive k (can be greater than n) such that A000005(n)/n = A000005(k)/k.

Original entry on oeis.org

2, 2, 2, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 1, 2, 2, 2, 3, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 1, 2, 2, 2, 2, 2, 2, 3, 1

Offset: 1

Tejo Vrush, Oct 04 2021

The first 1 occurs at a(4). The corresponding k value is 4.
The first 2 occurs at a(1). The corresponding k values are 1, 2.
The first 3 occurs at a(9). The corresponding k values are 9, 18, 24.
The first 4 occurs at a(243). The corresponding k values are 243, 405, 486, 810.
Does every number appear in this sequence?
The first 5 is at a(3189375) with k-values {3189375, 5467500, 6378750, 6561000, 8748000} while the first 6 is at 3176523 with k-values {3176523, 4084101, 6353046, 6806835, 8168202, 13613670}. If 7 appears it does so past 10^8. - Charles R Greathouse IV, Oct 05 2021

			For n = 9, the k such that A000005(9)/9 = 1/3 = A000005(k)/k are 9, 18, and 24. Therefore, a(9) = 3.

David A. Corneth, Table of n, a(n) for n = 1..10000
Charles R Greathouse IV, PARI/GP script

Cf. A000005, A090387, A090395, A005382, A005383, A348184.

Array[Function[r, Count[Range[Ceiling[4/r^2]], _?(DivisorSigma[0, #]/# == r &)]][DivisorSigma[0, #]/#] &, 105] (* or *)
Block[{nn = 7, m, s}, m = 2^(2 nn); s = KeySort@ PositionIndex[Array[DivisorSigma[0, #]/# &, m]]; s = Reverse@ KeyDrop[s, TakeWhile[Keys@ s, 4/#^2 > m &]]; Length /@ Array[Lookup[s, DivisorSigma[0, #]/#] &, 2^nn]] (* Michael De Vlieger, Oct 04 2021 *)

a(n) = {my(q=numdiv(n)/n); sum(i=1, 4/q^2, numdiv(i)/i == q);} \\ Michel Marcus, Oct 04 2021

a(n) = my(q=numdiv(n)/n, s=denominator(q), res = 0); forstep(i=s, 4/q^2, s, if(numdiv(i) == q * i, res++)); res \\ David A. Corneth, Oct 07 2021

PARI
```
\\ See Greathouse link
```

A352834 Least k > 0 such that denominator( d(kn)/(kn) ) = n, or 0 if no such k exists, where d = A000005 is the number-of-divisors function.

Original entry on oeis.org

1, 4, 1, 1, 1, 12, 1, 10, 12, 2, 1, 20, 1, 2, 1, 1, 1, 0, 1, 24, 1, 2, 1, 2, 1, 2, 1, 20, 1, 0, 1, 14, 1, 2, 1, 24, 1, 2, 1, 4, 1, 40, 1, 20, 12, 2, 1, 3, 1, 4, 1, 20, 1, 18, 1, 2, 1, 2, 1, 32, 1, 2, 12, 1, 1, 40, 1, 20, 1, 48, 1, 0, 1, 2, 36, 20, 1, 40, 1, 5, 1, 2, 1, 4, 1, 2, 1, 2, 1, 48

Offset: 1

M. F. Hasler, Apr 04 2022

nonn

This sequence is motivated by the fact that A091895(n) is always a multiple of n, so we list here the ratio A091895(n)/n.
Record values are a(1) = 1, a(2) = 4, a(6) = a(9) = 12, a(12) = 20,
  a(20) = a(36) = 24, a(42) = a(66) = 40, a(70) = a(90) = a(110) = a(120) =
  a(126) = a(130) = a(170) = a(190) = a(198) = 48, a(210) = a(330) = a(390) = 64,
  a(420) = a(660) = a(780) = a(900) = a(1020) = 96,
  a(1050) = a(1134) = 120, a(1470) = a(1680) = a(1890) = 144,
  a(2310) = a(2730) = a(3150) = a(3570) = a(3990) = a(4290) = 192,
  a(4320) = 210, a(6300) = 216, a(7560) = 240, a(9240) = a(10920) = 288,
  a(13860) = a(16380) = a(17820) = a(20020) = 336, a(20790) = 360,
  a(23760) = a(28080) = 420, a(34650) = a(40950) = 432,
  a(41580) = a(49140) = 480, a(60060) = a(78540) = a(80850) = a(87780) = 576,
  a(90090) = 672, ...
Up to n = 10^6, the terms are bounded by a(n) < 16*n^(1/3). The largest ratios r(n) := a(n)/n^(1/3) are r(2310) ~ 14.5, r(23760) ~ 14.6, r(60060) ~ 14.7, r(90090) ~ 14.99, r(154440) ~ 15.66, r(201960) = 14.3, r(270270) = 14.85, r(420420) = 14.4, r(510510) = 14.4, r(720720) = 14.05, ...

Cf. A000005 (number-of-divisors function), A090395 (denominator of d(n)/n), A091895 (a(n)*n), A091896 (indices of zeros of a(n)).

apply( {A352834(n,L=n^2*2)=forstep(k=n,L,n,denominator(numdiv(k)/k)==n&&return(k/n))}, [1..99])

a(n) = A091895(n)/n; a(n) = 0 iff n is in A091896.

Conjecture: a(n) = O(n^(1/3)).

A368523 Positive integers in decreasing order of tau(k)/k, where tau(k) = A000005(k).

Original entry on oeis.org

1, 2, 4, 3, 6, 8, 12, 5, 10, 9, 18, 24, 16, 20, 7, 14, 15, 30, 36, 28, 48, 40, 60, 21, 42, 32, 11, 22, 72, 13, 26, 27, 54, 56, 84, 44, 45, 90, 120, 80, 96, 33, 66, 25, 50, 17, 34, 52, 35, 70, 108, 64, 19, 38, 144, 39, 78, 180, 63, 126, 168, 88, 132, 100, 112

Offset: 1

Keith J. Bauer, Dec 28 2023

nonn

Because tau(k)/k is bounded by 2/sqrt(k), this sequence is well-defined.
In the case of ties, terms are sorted from least to greatest.
Let c be the j-th distinct value of tau(a(n))/a(n). Terms of this sequence for which tau(a(n))/a(n) >= c are the proper divisors of A059992(j + 1) for 1 <= j <= 4. True for j = 0 if the 0th value of c is taken to be infinity. Pattern breaks for j > 4.

			tau(1)/1 = tau(2)/2 = 1
tau(4)/4 = 3/4
tau(3)/3 = tau(6)/6 = 2/3
tau(8)/8 = tau(12)/12 = 1/2
tau(5)/5 = tau(10)/10 = 2/5
tau(9)/9 = tau(18)/18 = tau(24)/24 = 1/3

Cf. A000005, A090387, A090395, A323383, A354768.

length = 100
result = {}
for n = 1, length do
  local div_count = 0
  local root_n = math.sqrt(n)
  for d = 1, root_n do
    if n % d == 0 then
      div_count = div_count + 2
    end
  end
  if (root_n == math.floor(root_n)) then
    div_count = div_count - 1
  end
  result[n] = {n, div_count / n}
end
function compare(a, b)
  if a[2] ~= b[2] then
    return a[2] > b[2]
  else
    return a[1] < b[1]
  end
end
table.sort(result, compare)
i = 1
bound = 2 / math.sqrt(length)
while result[i][2] >= bound do
  io.write(result[i][1] .. ',')
  i = i + 1
end

nmax = 100; s = Sort[Table[{k, DivisorSigma[0, k]/k}, {k, 1, nmax^2}], #1[[2]] >= #2[[2]] &]; Table[s[[j, 1]], {j, 1, nmax}] (* Vaclav Kotesovec, Jan 04 2024 *)

A355140 n/d(n) rounded to the nearest integer, where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 4, 2, 3, 3, 6, 2, 7, 4, 4, 3, 9, 3, 10, 3, 5, 6, 12, 3, 8, 7, 7, 5, 15, 4, 16, 5, 8, 9, 9, 4, 19, 10, 10, 5, 21, 5, 22, 7, 8, 12, 24, 5, 16, 8, 13, 9, 27, 7, 14, 7, 14, 15, 30, 5, 31, 16, 11, 9, 16, 8, 34, 11, 17, 9, 36, 6, 37, 19, 13, 13, 19

Offset: 1

Sameer Khan, Jun 20 2022

nonn

In the ambiguous case, fractions are rounded up.

			a(1) = round (1 / 1) = 1;
a(4) = round (4 / 3) = 1;
a(5) = round (5 / 2) = 3;

Cf. A000005, A078709 (floor), A334762 (ceiling), A090395 (numerators), A090387 (denominators).

Table[Floor[n/DivisorSigma[0,n]+1/2],{n,100}] (* Harvey P. Dale, Dec 22 2022 *)

from sympy import divisor_count
def A355140(n): return (2*n+(d:=divisor_count(n)))//(2*d) # Chai Wah Wu, Jun 20 2022

a(n) = round (n / A000005(n)).

cp's OEIS Frontend

A353011 Indices of "late birds" in A090395 (denominator of d(n)/n): indices n such that A090395(k) > A090395(n) for all k > n.

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A090387 Numerator of d(n)/n, where d(n) (A000005) is the number of divisors of n.

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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.

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A091895 Least number k such that the denominator of d(k)/k = n, or zero if no such number exists, where d is the number-of-divisors function A000005.

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A348172 a(n) is the number of positive k (can be greater than n) such that A000005(n)/n = A000005(k)/k.

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A352834 Least k > 0 such that denominator( d(k*n)/(k*n) ) = n, or 0 if no such k exists, where d = A000005 is the number-of-divisors function.

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A368523 Positive integers in decreasing order of tau(k)/k, where tau(k) = A000005(k).

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A355140 n/d(n) rounded to the nearest integer, where d(n) is the number of divisors of n (A000005).

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A352834 Least k > 0 such that denominator( d(kn)/(kn) ) = n, or 0 if no such k exists, where d = A000005 is the number-of-divisors function.