A053640 -id:A053640 - cp's OEIS frontend

A038547 Least number with exactly n odd divisors.

1, 3, 9, 15, 81, 45, 729, 105, 225, 405, 59049, 315, 531441, 3645, 2025, 945, 43046721, 1575, 387420489, 2835, 18225, 295245, 31381059609, 3465, 50625, 2657205, 11025, 25515, 22876792454961, 14175, 205891132094649, 10395, 1476225, 215233605

Offset: 1

Tom Verhoeff

Also least odd number with exactly n divisors. - Lekraj Beedassy, Aug 30 2006
a(2n-1) = {1, 9, 81, 729, 225, 59049, ...} are the squares. A122842(n) = sqrt(a(2n-1)) = {1, 3, 9, 27, 15, 243, 729, 45, 6561, 19683, 135, 177147, 225, 105, 4782969, 14348907, 1215, ...}. - Alexander Adamchuk, Sep 13 2006
Also the least number k such that there are n partitions of k whose elements are consecutive integers. I.e., 1=1, 3=1+2=3, 9=2+3+4=4+5=9, 15=1+2+3+4+5=4+5+6=7+8=15, etc. - Robert G. Wilson v, Jun 02 2007
The politeness of an integer, A069283(n), is defined to be the number of its nontrivial runsum representations, and the sequence 3, 9, 15, 81, 45, 729, 105, ... represents the least integers to have a politeness of 1, 2, 3, 4, ... This is also the sequence of smallest integers with n+1 odd divisors and so apart from the leading 1, is precisely this sequence. - Ant King, Sep 23 2009
a(n) is also the least number k with the property that the symmetric representation of sigma(k) has n subparts. - Omar E. Pol, Dec 31 2016

			a(2^3) = 105 = 3*5 while a(2^4) = 945 = 3^3 * 5 * 7. There are 5 partition lists for the exponents of numbers with 16 odd divisors; they are {1, 1, 1, 1}, {3, 1, 1}, {3, 3}, {7, 1}, and {15} that result in the 5 numbers 1155, 945, 3375, 10935, and 14348907. Number a(3^8) = a(6561) = 3^2 * 5^2 * ... * 19^2 * 23^2 = 12442607161209225 while a(3^9) = a(19683) = 3^8 * 5^2 * ... * 19^2 * 23^2 = 9070660620521525025. The numbers a(5^52) = 3^4 * 5^4 * 7^4 * ... and a(5^53) = 3^24 * 5^4 * 7^4 * ... have 393 and 402 digits, respectively.  - _Hartmut F. W. Hoft_, Nov 03 2022

Don Reble, Table of n, a(n) for n = 1..2000
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

A122842 = sqrt( a(2n-1) ).
Cf. A001227, A005179, A002110, A070826, A000005, A053640, A053624, A237593, A279387.
Row 1 of A266531. - Omar E. Pol, Dec 31 2016
Cf. A006093, A076274, A080577.

import Data.List  (find)
import Data.Maybe (fromJust)
a038547 n = fromJust $ find ((== n) . length . divisors) [1,3..]
   where divisors m = filter ((== 0) . mod m) [1..m]
-- Reinhard Zumkeller, Feb 24 2011

Table[Select[Range[1,532000,2],DivisorSigma[0,#]==k+1 &,1],{k,0,15}]//Flatten (* Ant King, Nov 28 2010 *)
2#-1&/@With[{ds=DivisorSigma[0,Range[1,600000,2]]},Table[Position[ds,n,1,1],{n,16}]]//Flatten (* The program is not suitable for generating terms beyond a(16) *) (* Harvey P. Dale, Jun 06 2017 *)
(* direct computation of A038547(n) *)
(* Function by _Vaclav Kotesovec_in A005179, Apr 04 2021, modified for odd divisors *)
mp[1, m_] := {{}}; mp[n_, 1] := {{}}; mp[n_?PrimeQ, m_] := If[mHartmut F. W. Hoft, Mar 05 2023 *)

for(nd=1,15,forstep(k=1,10^66,2,if(nd==numdiv(k),print1(k,", ");break())))

from math import prod
from sympy import isprime, divisors, prime
def A038547(n):
    def mult_factors(n):
        if isprime(n):
            return [(n,)]
        c = []
        for d in divisors(n,generator=True):
            if 1Chai Wah Wu, Aug 17 2024

a(p) = 3^(p-1) for primes p. - Zak Seidov, Apr 18 2006
a(n) = A119265(n,n). - Reinhard Zumkeller, May 11 2006
It was suggested by Alexander Adamchuk that for all n >= 1, we have a(3^(n-1)) = (p(n)#/2)^2 = (A002110(n)/2)^2 = A070826(n)^2. But this is false! E.g., (p(n)#/2)^2 = 3^2 * 5^2 * 7^2 * ... * 23^2 * 29^2 does indeed have 3^9 odd factors, but it is greater than 3^8 * 5^2 * 7^2 * ... * 23^2 which has 9*3*3*3*3*3*3*3 = 9*3^7 = 3^9 odd factors. - Richard Sabey, Oct 06 2007
a(A053640(m)) = a(A000005(A053624(m))) = A053624(m). - Rick L. Shepherd, Apr 20 2008
a(p^k) = Product_{i=1..k} prime(i+1)^(p-1), p prime and k >= 0, only when p_(k+1) < 3^p. - Hartmut F. W. Hoft, Nov 03 2022

Corrected by Ron Knott, Feb 22 2001
a(30) from Zak Seidov, Apr 18 2006
a(32)-a(34) from Lekraj Beedassy, Aug 30 2006

A053624 Highly composite odd numbers: odd numbers where d(n) increases to a record.

Original entry on oeis.org

1, 3, 9, 15, 45, 105, 225, 315, 945, 1575, 2835, 3465, 10395, 17325, 31185, 45045, 121275, 135135, 225225, 405405, 675675, 1576575, 2027025, 2297295, 3828825, 6891885, 11486475, 26801775, 34459425, 43648605, 72747675, 130945815

Offset: 1

Stefano Lanfranco (lastefano(AT)yahoo.it), Mar 21 2000

Also numbers k such that the number of partitions of k into consecutive integers is a record. For example, 45 = 22+23 = 14+15+16 = 7+8+9+10+11 = 5+6+7+8+9+10 = 1+2+3+4+5+6+7+8+9, six such partitions, but all smaller terms have fewer such partitions (15 has four). See A000005 comments and A038547 formula. - Rick L. Shepherd, Apr 20 2008
From Hartmut F. W. Hoft, Mar 29 2022: (Start)
Also the odd parts of the numbers in A340506, see also comments in A250071.
A140864 is a subsequence. (End)
Positions of records in A001227, i.e., integers whose number of odd divisors sets a new record. - Bernard Schott, Jul 18 2022
Conjecture: all terms after the first three terms are congruent to 5 mod 10. - Harvey P. Dale, Jul 05 2023
From Keith F. Lynch, Jan 12 2024: (Start)
Dale's conjecture is correct.  a(n) can't be even, since then a(n)/2 would be a smaller number with the same number of odd divisors.  The respective powers of the successive odd primes can't increase, since if they did, swapping them would give a smaller number with the same number of divisors, e.g., 3^2 * 5^4 has the same number of divisors as 3^4 * 5^2, and the latter is smaller.  As such, every a(n) must be an odd multiple of 5, hence congruent to 5 mod 10, unless it's simply a power of 3.  But multiplying a power of 3 by 3 gives just one more divisor while multiplying a power of 3 by 5 doubles the number of divisors, so after a(n) = 9 all a(n) must be congruent to 5 mod 10, i.e., have a rightmost decimal digit of 5.
This has three equivalent definitions:
* Odd numbers with more divisors than any smaller odd number.
* Numbers with more odd divisors than any smaller number, i.e., record high values of A001227.
* Numbers with a greater excess of odd divisors over even divisors than any smaller number, i.e., record high values of A048272. (End)

			9 is in the sequence because 9 has 3 divisors {1, 3, 9}, which is more than any previous odd number.

Amiram Eldar, Table of n, a(n) for n = 1..363 (terms 1..170 from Ray Chandler)
David Ryan, Mathematical Harmony Analysis, arXiv preprint arXiv:1603.08904 [cs.SD], 2016.

Cf. A002182, A053640, A000005. Subsequence of A147516.
Cf. A140864, A237593, A250071, A279387, A340506.
Cf. A005408, A001227, A038547.

nn = 10^6; maxd = 0;
Reap[For[n = 1, n <= nn, n += 2, If[(nd = DivisorSigma[0, n]) > maxd, Print[n]; Sow[n]; maxd = nd]]][[2, 1]] (* Jean-François Alcover, Sep 20 2018, from PARI *)
next[n_] := Module[{k=n, r=DivisorSigma[0, n]}, While[DivisorSigma[0, k]<=r, k+=2]; k]
a053624[n_] := NestList[next, 1, n-1]/; n>=1 (* returns n numbers *)
a053624[31] (* Hartmut F. W. Hoft, Mar 29 2022 *)
DeleteDuplicates[Table[{n,DivisorSigma[0,n]},{n,1,131*10^6,2}],GreaterEqual[ #1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Jul 05 2023 *)

lista(nn) = {maxd = 0; forstep (n=1, nn, 2, if ((nd = numdiv(n)) > maxd, print1(n, ", "); maxd = nd;););} \\ Michel Marcus, Apr 21 2014

A340506 For those rows n of A249223 which are weakly increasing, let w(n) denote the maximal entry in the row: sequence gives values of n for which w(n) sets a new record.

Original entry on oeis.org

1, 6, 72, 120, 1440, 6720, 28800, 80640, 483840, 1612800, 5806080, 7096320, 85155840, 283852800, 510935040, 1476034560, 7947878400, 17712414720, 29520691200, 106274488320, 354248294400, 1653158707200, 2125489766400, 4817776803840, 8029628006400, 28906660823040

Offset: 1

N. J. A. Sloane, Jan 23 2021

This is a companion to A250071 (and is derived from the data for that sequence), which lists the first time k appears as a width.
The record values are 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, but more data is needed to identify this sequence.
The odd part of a(n) is A053624(n), n>=1. The record values 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 20, ... are the beginning of A053640. - Hartmut F. W. Hoft, Mar 29 2022

			a(4) = 120 = 2^3 * A053624(4) = 2^3 * 15 and a(7) = 28800 = 2^7 * A053624(7) = 2^7 * 225. - _Hartmut F. W. Hoft_, Mar 29 2022

Cf. A237270, A237593, A249223, A250068, A250070, A250071.

Cf. A053624, A053640.

prevPower2[k_] := If[k==1, 1, 2^(Ceiling[Log[2, k]]-1)]
a340506[n_] := Module[{recL={{1, 1}}, q, d, pp}, For[q=1, q<=n, q+=2, d=DivisorSigma[0, q]; pp=prevPower2[q] q; If[First[Last[recL]]Hartmut F. W. Hoft, Mar 29 2022 *)

a(n) = 2^t(n) * A053624(n), n > 1, where t(n) is the largest exponent satisfying 2^t(n) < A053624(n) and A053624(n) is the odd part of a(n) - see the comment in A250071. - Hartmut F. W. Hoft, Mar 29 2022

a(12)-a(26) from Hartmut F. W. Hoft, Mar 29 2022

A095717 "Second order" highly composite numbers: the gap between the number of divisors (d(n)) rises to a new record.

Original entry on oeis.org

2, 12, 120, 720, 2520, 5040, 110880, 1441440, 21621600, 367567200, 6983776800, 13967553600, 321253732800, 481880599200, 963761198400, 6746328388800, 55898149507200, 130429015516800, 195643523275200, 1732842634723200, 4043299481020800, 6064949221531200, 60649492215312000

Offset: 1

Stefano Lanfranco (lastefano(AT)yahoo.it), Jul 08 2004

The corresponding indices of the highly composite numbers are 2, 5, 10, 14, 18, 19, 30, 40, ... (see the link for more values). - Amiram Eldar, Jul 17 2019

			120 is in the sequence because d(120)=16 and the previous highly composite number is 60 with d(60)=12, the gap between the number of divisor 16-12=4 is the maximum with number <=120

Amiram Eldar, Table of n, a(n) for n = 1..1374
Amiram Eldar, Table of n and k such that A002182(k) = a(n) for n = 1..1374

Cf. A002182, A002183, A053640, A053624.

s={}; dmax = dmprev= gapmax=0; Do[d = DivisorSigma[0, k]; If[d > dmax ,  dmprev = dmax; dmax = d; gap = dmax - dmprev ;If[gap > gapmax, gapmax = gap; AppendTo[s, k]]], {k, 1, 1500000}]; s (* Amiram Eldar, Jul 17 2019 *)

Definition edited by Harvey P. Dale, Apr 09 2018

More terms from Amiram Eldar, Jul 17 2019

cp's OEIS Frontend

A038547 Least number with exactly n odd divisors.

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A053624 Highly composite odd numbers: odd numbers where d(n) increases to a record.

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A340506 For those rows n of A249223 which are weakly increasing, let w(n) denote the maximal entry in the row: sequence gives values of n for which w(n) sets a new record.

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Programs

Mathematica

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A095717 "Second order" highly composite numbers: the gap between the number of divisors (d(n)) rises to a new record.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions