A038547 -id:A038547 - cp's OEIS frontend

A122812 Numbers k where A046523(A005179(k)) differs from A046523(A038547(k)).

8, 24, 48, 64, 72, 80, 108, 112, 128, 144, 160, 162, 176, 192, 208, 216, 224, 243, 256, 272, 288, 304, 320, 324, 352, 368, 384, 416, 432, 448, 464, 480, 486, 496, 512, 544, 576, 592, 608, 640, 648, 656, 672, 688, 704, 729, 736, 752, 768, 832, 848, 864, 896

Offset: 1

Ray Chandler, Sep 22 2006

nonn

Where the prime signature of the least number with exactly k divisors differs from the prime signature of the least odd number with exactly k divisors.
Also where A122375(k) differs from A122810(k).
Also where A122376(k) differs from A122811(k).

Amiram Eldar, Table of n, a(n) for n = 1..2000

Cf. A005179, A038547, A046523, A122375, A122376, A122810-A122819.

A122842 Square roots of the odd indexed terms of A038547.

Original entry on oeis.org

1, 3, 9, 27, 15, 243, 729, 45, 6561, 19683, 135, 177147, 225, 105, 4782969, 14348907, 1215, 675, 387420489, 3645, 3486784401, 10460353203, 315, 94143178827, 3375, 32805, 2541865828329, 6075, 98415, 68630377364883

Offset: 1

Alexander Adamchuk, Sep 13 2006, Sep 25 2006

nonn

A038547(n) is the least number with exactly n odd divisors. For odd n these are perfect squares.

			a(1) = 1 because A038547(1) = 1.
a(2) = 3 because A038547(3) = 9.
a(5) = 15 because A038547(9) = 225.

Amiram Eldar, Table of n, a(n) for n = 1..2100 (terms 1..1000 from R. J. Mathar)

Cf. A038547, A121858, A001227, A005179, A119265.

Cf. A007051, A070826, A002110.

(* Function a038547[ ] is defined in A038547 *)
a122842[n_]:=Sqrt[a038547[2n-1]]
Map[a122842,Range[30]] (* Hartmut F. W. Hoft, Feb 07 2023 *)

a(n) = sqrt(A038547(2*n-1)).
a(n) = sqrt(A119265(2*n-1, 2*n-1)).
a(n) = 3^(n-1) for n = (p+1)/2, where p is an odd prime.
a(3*n+2) = 5*3^n for n = (p-1)/2, where p is an odd prime.

More terms from R. J. Mathar, Sep 20 2006

Edited by Hartmut F. W. Hoft, Feb 07 2023

A119265 Triangle read by rows, 1<=k<=n: T(n,k) = k-th divisor of the smallest odd number with exactly n divisors, A038547.

Original entry on oeis.org

1, 1, 3, 1, 3, 9, 1, 3, 5, 15, 1, 3, 9, 27, 81, 1, 3, 5, 9, 15, 45, 1, 3, 9, 27, 81, 243, 729, 1, 3, 5, 7, 15, 21, 35, 105, 1, 3, 5, 9, 15, 25, 45, 75, 225, 1, 3, 5, 9, 15, 27, 45, 81, 135, 405, 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, 1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105

Offset: 1

Reinhard Zumkeller, May 11 2006

Jean-François Alcover, Table of n, a(n) for n = 1..5050

Cf. A081532, A000005, A005408.

A038547 = Cases[Import["https://oeis.org/A038547/b038547.txt", "Table"], {, }][[All, 2]];
row[n_] := row[n] = Divisors[A038547[[n]]];
T[n_, k_] := row[n][[k]];
Table[T[n, k], {n, 1, 100}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 19 2021 *)

T(n,1) = 1; T(n,n) = A038547(n);

T(n,k) = A027750(A006218(A038547(n)-1) + k).

A122813 Terms in A005179 where prime signature differs from that of corresponding term in A038547.

Original entry on oeis.org

24, 360, 2520, 7560, 10080, 15120, 50400, 60480, 83160, 110880, 166320, 352800, 967680, 332640, 3870720, 554400, 665280, 2822400, 1081080, 61931520, 1441440, 247726080, 2162160, 3880800, 10644480, 3963617280, 4324320, 42577920, 7207200

Offset: 1

Ray Chandler, Sep 22 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1371

Cf. A005179, A038547, A046523, A122812, A122814.

a(n) = A005179(A122812(n)).

A122814 Terms in A038547 where prime signature differs from that of corresponding term in A005179.

Original entry on oeis.org

105, 3465, 45045, 135135, 225225, 405405, 1576575, 3648645, 2297295, 3828825, 6891885, 17342325, 295540245, 11486475, 2659862205, 26801775, 62026965, 225450225, 43648605, 215448838605, 72747675, 1939039547445, 130945815

Offset: 1

Ray Chandler, Sep 22 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..932

Cf. A005179, A038547, A046523, A122812, A122813.

a(n) = A038547(A122812(n)).

A360435 a(n) = A038547(3^n), smallest number with 3^n odd divisors.

Original entry on oeis.org

9, 225, 11025, 1334025, 225450225, 65155115025, 23520996524025, 12442607161209225, 9070660620521525025, 7628425581858602546025, 7330916984166117046730025, 10036025351323414236973404225, 16870558615574659332352292502225, 31193662880197545105519388836614025, 68906801302356377138092329940080381225

Offset: 1

Hartmut F. W. Hoft, Feb 07 2023

nonn

			a(3) = A038547(3^3) = 11025 = 3^2 * 5^2 * 7^2,
a(8) = A038547(3^8) = 12442607161209225 = 3^2 * ... * 23^2,
a(9) = A038547(3^9) = 9070660620521525025 = 3^8 * 5^2 * ... * 23^2 since 23^2 = 529 < 3^6 = 729 < 29^2 = 841,
a(10) = A038547(3^10) = 7628425581858602546025 = 3^8 * 5^2 * ... * 29^2.
a(31) = A038547(3^31) = 3^8 * 5^8 * 7^2 * ... * 113^2.
a(70) = A038547(3^70) = 3^8 * 5^8 * 7^8 * 11^2 * ... * 337^2.

Cf. A001227, A005179, A007051, A038547, A070826, A122842.

value3[part_] := Module[{len=Length[part]}, Apply[Times, Map[#[[1]]^(3^#[[2]]-1)&, Transpose[{Map[Prime, Range[2, len+1]], part}]]]]
a360435[n_] := Module[{pL=Reverse[IntegerPartitions[n]], min, i=2, next}, min=value3[pL[[1]]]; While[i<=Length[pL]&&3^(3^pL[[i, 1]]-1)

a(n)={my(m=vecprod(primes(n+1))^2/4, b=logint(logint(m,3)+1,3)); forpart(p=n, m=min(m, prod(i=1, #p, prime(1+i)^(3^p[#p+1-i]-1))), [1, b]); m} \\ Andrew Howroyd, Feb 07 2023

a(prime(n) - 1) = A038547(A038547(prime(n))), n >= 1.

A001227 Number of odd divisors of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 4, 1, 2, 3, 2, 2, 4, 2, 2, 2, 3, 2, 4, 2, 2, 4, 2, 1, 4, 2, 4, 3, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 2, 2, 3, 3, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 6, 1, 4, 4, 2, 2, 4, 4, 2, 3, 2, 2, 6, 2, 4, 4, 2, 2, 5, 2, 2, 4, 4, 2, 4, 2, 2, 6, 4, 2, 4, 2, 4, 2, 2, 3, 6, 3, 2, 4, 2, 2, 8

Offset: 1

N. J. A. Sloane

Also (1) number of ways to write n as difference of two triangular numbers (A000217), see A136107; (2) number of ways to arrange n identical objects in a trapezoid. - Tom Verhoeff
Also number of partitions of n into consecutive positive integers including the trivial partition of length 1 (e.g., 9 = 2+3+4 or 4+5 or 9 so a(9)=3). (Useful for cribbage players.) See A069283. - Henry Bottomley, Apr 13 2000
This has been described as Sylvester's theorem, but to reduce ambiguity I suggest calling it Sylvester's enumeration. - Gus Wiseman, Oct 04 2022
a(n) is also the number of factors in the factorization of the Chebyshev polynomial of the first kind T_n(x). - Yuval Dekel (dekelyuval(AT)hotmail.com), Aug 28 2003
Number of factors in the factorization of the polynomial x^n+1 over the integers. See also A000005. - T. D. Noe, Apr 16 2003
a(n) = 1 if and only if n is a power of 2 (see A000079). - Lekraj Beedassy, Apr 12 2005
Number of occurrences of n in A049777. - Philippe Deléham, Jun 19 2005
For n odd, n is prime if and only if a(n) = 2. - George J. Schaeffer (gschaeff(AT)andrew.cmu.edu), Sep 10 2005
Also number of partitions of n such that if k is the largest part, then each of the parts 1,2,...,k-1 occurs exactly once. Example: a(9)=3 because we have [3,3,2,1],[2,2,2,2,1] and [1,1,1,1,1,1,1,1,1]. - Emeric Deutsch, Mar 07 2006
Also the number of factors of the n-th Lucas polynomial. - T. D. Noe, Mar 09 2006
Lengths of rows of triangle A182469;
Denoted by Delta_0(n) in Glaisher 1907. - Michael Somos, May 17 2013
Also the number of partitions p of n into distinct parts such that max(p) - min(p) < length(p). - Clark Kimberling, Apr 18 2014
Row sums of triangle A247795. - Reinhard Zumkeller, Sep 28 2014
Row sums of triangle A237048. - Omar E. Pol, Oct 24 2014
A069288(n) <= a(n). - Reinhard Zumkeller, Apr 05 2015
A000203, A000593 and this sequence have the same parity: A053866. - Omar E. Pol, May 14 2016
a(n) is equal to the number of ways to write 2*n-1 as (4*x + 2)*y + 4*x + 1 where x and y are nonnegative integers. Also a(n) is equal to the number of distinct values of k such that k/(2*n-1) + k divides (k/(2*n-1))^(k/(2*n-1)) + k, (k/(2*n-1))^k + k/(2*n-1) and k^(k/(2*n-1)) + k/(2*n-1). - Juri-Stepan Gerasimov, May 23 2016, Jul 15 2016
Also the number of odd divisors of n*2^m for m >= 0. - Juri-Stepan Gerasimov, Jul 15 2016
a(n) is odd if and only if n is a square or twice a square. - Juri-Stepan Gerasimov, Jul 17 2016
a(n) is also the number of subparts in the symmetric representation of sigma(n). For more information see A279387 and A237593. - Omar E. Pol, Nov 05 2016
a(n) is also the number of partitions of n into an odd number of equal parts. - Omar E. Pol, May 14 2017 [This follows from the g.f. Sum_{k >= 1} x^k/(1-x^(2*k)). - N. J. A. Sloane, Dec 03 2020]

			G.f. = q + q^2 + 2*q^3 + q^4 + 2*q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + 2*q^10 + ...
From _Omar E. Pol_, Nov 30 2020: (Start)
For n = 9 there are three odd divisors of 9; they are [1, 3, 9]. On the other hand there are three partitions of 9 into consecutive parts: they are [9], [5, 4] and [4, 3, 2], so a(9) = 3.
Illustration of initial terms:
                              Diagram
   n   a(n)                         _
   1     1                        _|1|
   2     1                      _|1 _|
   3     2                    _|1  |1|
   4     1                  _|1   _| |
   5     2                _|1    |1 _|
   6     2              _|1     _| |1|
   7     2            _|1      |1  | |
   8     1          _|1       _|  _| |
   9     3        _|1        |1  |1 _|
  10     2      _|1         _|   | |1|
  11     2    _|1          |1   _| | |
  12     2   |1            |   |1  | |
...
a(n) is the number of horizontal line segments in the n-th level of the diagram. For more information see A286001. (End)

B. C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, see p. 487 Entry 47.
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 306.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4).
Ronald. L. Graham, Donald E. Knuth, and Oren Patashnik, Concrete Mathematics, 2nd ed. (Addison-Wesley, 1994), see exercise 2.30 on p. 65.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 28.

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
K. S. Brown's Mathpages, Partitions into Consecutive Integers.
Atli Fannar Franklín, Pattern avoidance enumerated by inversions, arXiv:2410.07467 [math.CO], 2024. See pp. 2, 18.
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
A. Heiligenbrunner, Sum of adjacent numbers (in German).
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 11.
Gerzson Kéri, The factorization of compressed Chebyshev polynomials and other polynomials related to multiple-angle formulas, Annales Univ. Sci. Budapest (Hungary, 2022) Sect. Comp., Vol. 53, 93-108.
Mircea Merca, Combinatorial interpretations of a recent convolution for the number of divisors of a positive integer, Journal of Number Theory, Volume 160, March 2016, Pages 60-75, function tau_o(n).
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.
R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976.
N. J. A. Sloane, Transforms.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article 99.1.6.
Eric Weisstein's World of Mathematics, Binomial Number and Odd Divisor Function.
Eric Weisstein's World of Mathematics, q-Polygamma Function.
Index entries for "core" sequences.
Index entries for sequences mentioned by Glaisher.

Cf. A000005, A000079, A000593, A010054 (char. func.), A038547 (positions of first appearances), A050999, A051000, A051001, A051002, A051731, A054844, A069283, A069288, A109814, A115369, A118235, A118236, A125911, A136655, A183063, A183064, A237593, A247795, A272887, A273401, A279387, A286001.
Cf. A000203, A001620, A002162, A053866, A060831.
If this sequence counts gapless sets by sum (by Sylvester's enumeration), these sets are ranked by A073485 and A356956. See also A055932, A066311, A073491, A107428, A137921, A333217, A356224, A356841, A356845.
Dirichlet inverse is A327276.

a001227 = sum . a247795_row
-- Reinhard Zumkeller, Sep 28 2014, May 01 2012, Jul 25 2011

[NumberOfDivisors(n)/Valuation(2*n, 2): n in [1..100]]; // Vincenzo Librandi, Jun 02 2019

for n from 1 by 1 to 100 do s := 0: for d from 1 by 2 to n do if n mod d = 0 then s := s+1: fi: od: print(s); od:
A001227 := proc(n) local a,d;
    a := 1 ;
    for d in ifactors(n)[2] do
        if op(1,d) > 2 then
            a := a*(op(2,d)+1) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jun 18 2015

f[n_] := Block[{d = Divisors[n]}, Count[ OddQ[d], True]]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Aug 27 2004 *)
Table[Total[Mod[Divisors[n], 2]],{n,105}] (* Zak Seidov, Apr 16 2010 *)
f[n_] := Block[{d = DivisorSigma[0, n]}, If[ OddQ@ n, d, d - DivisorSigma[0, n/2]]]; Array[f, 105] (* Robert G. Wilson v *)
a[ n_] := Sum[  Mod[ d, 2], { d, Divisors[ n]}]; (* Michael Somos, May 17 2013 *)
a[ n_] := DivisorSum[ n, Mod[ #, 2] &]; (* Michael Somos, May 17 2013 *)
Count[Divisors[#],?OddQ]&/@Range[110] (* _Harvey P. Dale, Feb 15 2015 *)
(* using a262045 from A262045 to compute a(n) = number of subparts in the symmetric representation of sigma(n) *)
(* cl = current level, cs = current subparts count *)
a001227[n_] := Module[{cs=0, cl=0, i, wL, k}, wL=a262045[n]; k=Length[wL]; For[i=1, i<=k, i++, If[wL[[i]]>cl, cs++; cl++]; If[wL[[i]]Hartmut F. W. Hoft, Dec 16 2016 *)
a[n_] := DivisorSigma[0, n / 2^IntegerExponent[n, 2]]; Array[a, 100] (* Amiram Eldar, Jun 12 2022 *)

{a(n) = sumdiv(n, d, d%2)}; /* Michael Somos, Oct 06 2007 */

{a(n) = direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( 4, p) * X))[n]}; /* Michael Somos, Oct 06 2007 */

a(n)=numdiv(n>>valuation(n,2)) \\ Charles R Greathouse IV, Mar 16 2011

a(n)=sum(k=1,round(solve(x=1,n,x*(x+1)/2-n)),(k^2-k+2*n)%(2*k)==0) \\ Charles R Greathouse IV, May 31 2013

a(n)=sumdivmult(n,d,d%2) \\ Charles R Greathouse IV, Aug 29 2013

from functools import reduce
from operator import mul
from sympy import factorint
def A001227(n): return reduce(mul,(q+1 for p, q in factorint(n).items() if p > 2),1) # Chai Wah Wu, Mar 08 2021

def A001227(n): return len([1 for d in divisors(n) if is_odd(d)])
[A001227(n) for n in (1..80)]  # Peter Luschny, Feb 01 2012

Dirichlet g.f.: zeta(s)^2*(1-1/2^s).
Comment from N. J. A. Sloane, Dec 02 2020: (Start)
By counting the odd divisors f n in different ways, we get three different ways of writing the ordinary generating function.  It is:
A(x) =  x + x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + x^8 + 3*x^9 + 2*x^10 + ...
  = Sum_{k >= 1} x^(2*k-1)/(1-x^(2*k-1))
  = Sum_{k >= 1} x^k/(1-x^(2*k))
  = Sum_{k >= 1} x^(k*(k+1)/2)/(1-x^k)  [Ramanujan, 2nd notebook, p. 355.].
(This incorporates comments from Vladeta Jovovic, Oct 16 2002 and  Michael Somos, Oct 30 2005.) (End)
G.f.: x/(1-x) + Sum_{n>=1} x^(3*n)/(1-x^(2*n)), also L(x)-L(x^2) where L(x) = Sum_{n>=1} x^n/(1-x^n). - Joerg Arndt, Nov 06 2010
a(n) = A000005(n)/(A007814(n)+1) = A000005(n)/A001511(n).
Multiplicative with a(p^e) = 1 if p = 2; e+1 if p > 2. - David W. Wilson, Aug 01 2001
a(n) = A000005(A000265(n)). - Lekraj Beedassy, Jan 07 2005
Moebius transform is period 2 sequence [1, 0, ...] = A000035, which means a(n) is the Dirichlet convolution of A000035 and A057427.
a(n) = A113414(2*n). - N. J. A. Sloane, Jan 24 2006 (corrected Nov 10 2007)
a(n) = A001826(n) + A001842(n). - Reinhard Zumkeller, Apr 18 2006
Sequence = M*V = A115369 * A000005, where M = an infinite lower triangular matrix and V = A000005, d(n); as a vector: [1, 2, 2, 3, 2, 4, ...]. - Gary W. Adamson, Apr 15 2007
Equals A051731 * [1,0,1,0,1,...]; where A051731 is the inverse Mobius transform. - Gary W. Adamson, Nov 06 2007
a(n) = A000005(n) - A183063(n).
a(n) = d(n) if n is odd, or d(n) - d(n/2) if n is even, where d(n) is the number of divisors of n (A000005). (See the Weisstein page.) - Gary W. Adamson, Mar 15 2011
Dirichlet convolution of A000005 and A154955 (interpreted as a flat sequence). - R. J. Mathar, Jun 28 2011
a(A000079(n)) = 1; a(A057716(n)) > 1; a(A093641(n)) <= 2; a(A038550(n)) = 2; a(A105441(n)) > 2; a(A072502(n)) = 3. - Reinhard Zumkeller, May 01 2012
a(n) = 1 + A069283(n). - R. J. Mathar, Jun 18 2015
a(A002110(n)/2) = n, n >= 1. - Altug Alkan, Sep 29 2015
a(n*2^m) = a(n*2^i), a((2*j+1)^n) = n+1 for m >= 0, i >= 0 and j >= 0. a((2*x+1)^n) = a((2*y+1)^n) for positive x and y. - Juri-Stepan Gerasimov, Jul 17 2016
Conjectures: a(n) = A067742(n) + 2*A131576(n) = A082647(n) + A131576(n). - Omar E. Pol, Feb 15 2017
a(n) = A000005(2n) - A000005(n) = A099777(n)-A000005(n). - Danny Rorabaugh, Oct 03 2017
L.g.f.: -log(Product_{k>=1} (1 - x^(2*k-1))^(1/(2*k-1))) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, Jul 30 2018
G.f.: (psi_{q^2}(1/2) + log(1-q^2))/log(q), where psi_q(z) is the q-digamma function. - Michael Somos, Jun 01 2019
a(n) = A003056(n) - A238005(n). - Omar E. Pol, Sep 12 2021
Sum_{k=1..n} a(k) ~ n*log(n)/2 + (gamma + log(2)/2 - 1/2)*n, where gamma is Euler's constant (A001620). - Amiram Eldar, Nov 27 2022
Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A000005(k) = log(2) (A002162). - Amiram Eldar, Mar 01 2023
a(n) = Sum_{i=1..n} (-1)^(i+1)*A135539(n,i). - Ridouane Oudra, Apr 13 2023

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

A303555 Triangle read by rows: T(n,k) = 2^(n-k)*prime(k)#, 1 <= k <= n, where prime(k)# is the product of first k primes.

Original entry on oeis.org

2, 4, 6, 8, 12, 30, 16, 24, 60, 210, 32, 48, 120, 420, 2310, 64, 96, 240, 840, 4620, 30030, 128, 192, 480, 1680, 9240, 60060, 510510, 256, 384, 960, 3360, 18480, 120120, 1021020, 9699690, 512, 768, 1920, 6720, 36960, 240240, 2042040, 19399380, 223092870, 1024, 1536, 3840, 13440, 73920, 480480, 4084080, 38798760, 446185740, 6469693230

Offset: 1

Ilya Gutkovskiy, Apr 26 2018

T(n,k) = the smallest number m having exactly n prime divisors counted with multiplicity and exactly k distinct prime divisors.

			T(5,4) = 420 = 2^2*3*5*7, hence 420 is the smallest number m such that bigomega(m) = 5 and omega(m) = 4 (see A189982).
Triangle begins:
    2;
    4,   6;
    8,  12,  30;
   16,  24,  60,  210;
   32,  48, 120,  420, 2310;
   64,  96, 240,  840, 4620, 30030;
  128, 192, 480, 1680, 9240, 60060, 510510;
  ...

Eric Weisstein's World of Mathematics, Prime Factor
Eric Weisstein's World of Mathematics, Distinct Prime Factors
Eric Weisstein's World of Mathematics, Primorial
Index entries for sequences related to primorial numbers
Index to sequences related to prime signature

Cf. A000079, A001221, A001222, A002110, A005179, A038547, A055079, A070175, A303557 (central terms).

Flatten[Table[2^(n - k) Product[Prime[j], {j, k}], {n, 10}, {k, n}]]

A072502 Numbers that are run sums (trapezoidal, the difference between two triangular numbers) in exactly 3 ways.

Original entry on oeis.org

9, 18, 25, 36, 49, 50, 72, 98, 100, 121, 144, 169, 196, 200, 242, 288, 289, 338, 361, 392, 400, 484, 529, 576, 578, 676, 722, 784, 800, 841, 961, 968, 1058, 1152, 1156, 1352, 1369, 1444, 1568, 1600, 1681, 1682, 1849, 1922, 1936, 2116, 2209, 2304, 2312, 2704

Offset: 1

Ron Knott, Jan 27 2003

Also numbers that are the product of a power of 2 (A000079) and the square of an odd prime, or numbers having exactly 3 odd divisors: A001227(a(n)) = 3. - Reinhard Zumkeller, May 01 2012
Numbers n such that the symmetric representation of sigma(n) has 3 subparts. - Omar E. Pol, Dec 28 2016
Also numbers that can be expressed as the sum of k > 1 consecutive positive integers in exactly 2 ways. E.g., 2+3+4 = 9 and 4+5 = 9, 3+4+5+6 = 18 and 5+6+7 = 18. - Julie Jones, Aug 13 2018
Appears to be numbers n such that tau(2*n) = tau(n) + 3. - Gary Detlefs, Jan 22 2020
Column 3 of A266531. - Omar E. Pol, Dec 01 2020

			a(1)=9 is the smallest number with 3 run sums: 2+3+4 = 4+5 = 9.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ron Knott, Introducing Runsums - a sum of consecutive integers.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2 (1999), Article #99.1.6.

Not to be confused with A069562.

Cf. A001227, A001248, A038547, A038550, A085548, A237593, A266531, A279387.

import Data.Set (singleton, deleteFindMin, insert)
a072502 n = a072502_list !! (n-1)
a072502_list = f (singleton 9) $ drop 2 a001248_list where
   f s (x:xs) = m : f (insert (2 * m) $ insert x s') xs where
                (m,s') = deleteFindMin s
-- Reinhard Zumkeller, May 01 2012

Sum_{n>=1} 1/a(n) = 2 * Sum_{p odd prime} 1/p^2 = 2 * A085548 - 1/2 = 0.404494... - Amiram Eldar, Feb 18 2021

Extended by Ray Chandler, Dec 30 2011

cp's OEIS Frontend

A122812 Numbers k where A046523(A005179(k)) differs from A046523(A038547(k)).

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A122842 Square roots of the odd indexed terms of A038547.

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A119265 Triangle read by rows, 1<=k<=n: T(n,k) = k-th divisor of the smallest odd number with exactly n divisors, A038547.

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A122813 Terms in A005179 where prime signature differs from that of corresponding term in A038547.

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A122814 Terms in A038547 where prime signature differs from that of corresponding term in A005179.

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A360435 a(n) = A038547(3^n), smallest number with 3^n odd divisors.

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A001227 Number of odd divisors of n.

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

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