A091896 -id:A091896 - 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

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

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)).

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

Original entry on oeis.org

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).

A353320 Odd numbers m such that there exists no k for which the denominator of d(k)/k = m where d(k) is the number of divisors of k (A000005).

Original entry on oeis.org

243, 625, 1875, 3969, 4375, 5625, 6875, 8125, 9801, 10625, 11875, 13125, 13689, 14375, 18125, 19375, 19845, 20625, 23125, 23409, 24375, 25625, 26875, 29241, 29375, 30625, 31875, 33125, 35625, 36875, 38125, 41875, 42849, 43125, 43659, 44375, 45625, 49375, 50625, 51597

Offset: 1

David A. Corneth, Apr 11 2022

nonn

Cf. A000005, A091896.

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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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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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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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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A353320 Odd numbers m such that there exists no k for which the denominator of d(k)/k = m where d(k) is the number of divisors of k (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.