A036484 -id:A036484 - cp's OEIS frontend

A036493 Largest number having binary order n (A029837) and of which the number of divisors is maximal in that range of g(k) = n.

1, 2, 4, 8, 12, 30, 60, 120, 240, 504, 840, 1680, 3960, 7560, 15120, 32760, 65520, 131040, 262080, 498960, 997920, 1965600, 3603600, 7207200, 14414400, 32432400, 64864800, 122522400, 245044800, 514594080, 1029188160, 2095133040, 4227022800, 8454045600

Offset: 0

Labos Elemer

nonn

This sequence differs from A036451 only at n = 3, 5, 9, 12, and 15, which are the values of n for which there exists more than one k such that g(k) = n and d(k) has the maximum possible value.

a(n) is the largest term k in A067128 such that log_2(k) <= n. - Jon E. Schoenfield, May 13 2018

			For n = 9, k is in {257, 512}, max(d(k)) = 24 (see A036451); this holds for four different numbers (360, 420, 480, and 504); a(9) = 504 since it is the largest.

Cf. A000005, A029837, A005179, A007416, A036451, A036470, A036484.

{1}~Join~Table[Max@ MaximalBy[Range[2^(n - 1) + 1, 2^n], DivisorSigma[0, #] &], {n, 24}] (* Michael De Vlieger, Aug 01 2017 *)

a(22)-a(24) from Michael De Vlieger, Aug 01 2017

a(25)-a(33) from Jon E. Schoenfield, May 12 2018

A346729 Maximum number of divisors among n-bit numbers.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96, 120, 144, 168, 200, 240, 288, 360, 432, 504, 600, 720, 864, 1008, 1152, 1344, 1600, 1920, 2304, 2688, 3072, 3584, 4096, 4800, 5760, 6720, 7680, 8640, 10080, 11520, 13824, 16128, 18432, 20736, 23040, 27648

Offset: 1

Jon E. Schoenfield, Jul 30 2021

nonn

a(n) is the maximum value of tau(k)=A000005(k) for k in the interval [2^(n-1), 2^n - 1]. For n >= 3, that smallest k at which tau(k) is maximized in that interval is A036484(n).

No term is repeated: for n >= 1, if k is the number in [2^(n-1), 2^n - 1] at which tau(k) is maximized (i.e., tau(k) = a(n)), then 2k, which will be a number in [2^n, 2^(n+1) - 1], will have more divisors than k has, so a(n+1) >= tau(2k) > tau(k) = a(n).

			There are four 3-bit numbers: 4 = 100_2, 5 = 101_2 = 5, 6 = 110_2, 7 = 111_2. 5 and 7 are both prime, so each has 2 divisors; 4 = 2^2 has 3 divisors (1, 2, and 4), and 6 = 2*3 has 4 divisors (1, 2, 3, and 6). Thus, the maximum number of divisors among 3-bit numbers is A000005(6) = 4, so a(3)=4.

Cf. A000005, A002183, A036484, A346730.

a[n_]:=Max[Table[DivisorSigma[0,k],{k,2^(n-1),2^n-1}]]; Table[a[n],{n,23}] (* Stefano Spezia, Aug 02 2021 *)

a(n) = vecmax(apply(numdiv, [2^(n-1)..2^n-1])); \\ Michel Marcus, Aug 03 2021

from sympy import divisors
def a(n): return max(len(divisors(n)) for n in range(2**(n-1), 2**n))
print([a(n) for n in range(1, 18)]) # Michael S. Branicky, Aug 02 2021

cp's OEIS Frontend

A036493 Largest number having binary order n (A029837) and of which the number of divisors is maximal in that range of g(k) = n.

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