A036470 -id:A036470

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A036451

Maximal value of d(x) (the number of divisors of x, A000005) if the binary order (see A029837) of x, the value g(x) = n.

+10
5

1, 2, 3, 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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

g(x) <= n can be replaced by g(x) = n.

LINKS

Giovanni Resta, Table of n, a(n) for n = 0..250 (based on A002182 b-file)

EXAMPLE

In the range of g(x) <= 5, the values of d(x) can be: 1, 2, 3, 4, 5, 6, 8 of which 8 is the maximal, so a(n) = a(g(x)) = 8.

MATHEMATICA

Max /@ Table[DivisorSigma[0, Floor[2^(n - 1) + k]], {n, 0, 22}, {k, Ceiling[2^(n - 1)]}] (* Michael De Vlieger, May 10 2017 *)

CROSSREFS

Cf. A000005, A002182, A029837, A005179, A007416, A036470-A036472.

KEYWORD

nonn

AUTHOR

Labos Elemer

EXTENSIONS

a(22)-a(32) from Alex Ratushnyak, Jun 06 2013

a(33)-a(49) from Giovanni Resta, Jun 06 2013

STATUS

approved

A036484

a(n) is the minimal number of binary order n which has maximal number of divisors in this interval.

+10
2

1, 2, 4, 6, 12, 24, 60, 120, 240, 360, 840, 1680, 2520, 7560, 15120, 27720, 55440, 110880, 221760, 498960, 720720, 1441440, 3603600, 7207200, 14414400, 32432400, 61261200, 122522400, 245044800, 367567200, 735134400, 2095133040

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Compare with A007416, where terms of this sequence are present.

LINKS

Table of n, a(n) for n=0..31.

EXAMPLE

For n=9, with 256 < k <= 512, d(k) takes 17 distinct values, of which d(k)=24 is the greatest (see A036451 and A036470) and occurs first at k=360, so a(9)=360.

MATHEMATICA

Block[{nn = 22, s}, s = TakeList[Array[DivisorSigma[0, # + 1] &, 2^nn - 1], 2^Range[0, nn - 1]]; {1}~Join~Map[2^(#1 - 1) + #2 & @@ FirstPosition[s, #] &, Map[Max, s]]] (* Michael De Vlieger, Nov 04 2020 *)

CROSSREFS

Cf. A000005, A029837, A005179, A007416, A036470, A036492, A036493.

KEYWORD

nonn,more

AUTHOR

Labos Elemer

EXTENSIONS

a(22)-a(31) from Sean A. Irvine, Nov 04 2020

STATUS

approved

A036493

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

+10
2

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..33.

EXAMPLE

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.

MATHEMATICA

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

CROSSREFS

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

KEYWORD

nonn

AUTHOR

Labos Elemer

EXTENSIONS

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

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

STATUS

approved

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