A060775 The greatest divisor d|n such that d < n/d, with a(1) = 1.

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 4, 3, 2, 1, 4, 1, 2, 3, 4, 1, 5, 1, 4, 3, 2, 5, 4, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 6, 1, 5, 3, 4, 1, 6, 5, 7, 3, 2, 1, 6, 1, 2, 7, 4, 5, 6, 1, 4, 3, 7, 1, 8, 1, 2, 5, 4, 7, 6, 1, 8, 3, 2, 1, 7, 5, 2, 3

Offset

1,6

Comments

Also: Largest divisor of n which is less than sqrt(n).

If n is not a square, then a(n) = A033676(n), else a(n) is strictly smaller than A033676(n) = sqrt(n) (except for a(1) = 1). - M. F. Hasler, Sep 20 2011

Record values occur for n = k * (k+1), for which a(n) = k. - Franklin T. Adams-Watters, May 01 2015

If we define a divisor d|n to be strictly inferior if d < n/d, then strictly inferior divisors are counted by A056924 and listed by A341674. This sequence gives the greatest strictly inferior divisor, which may differ from the lower central divisor A033676. Central divisors are listed by A207375. - Gus Wiseman, Feb 28 2021

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms n = 2..1000 from Harry J. Smith)

Formula

a(n) = max { d: d|n and d < sqrt(n) or d = 1 }, where "|" means "divides". [Corrected by M. F. Hasler, Apr 03 2019]

Example

n = 252, D = {1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252}, 18 divisors, the 9th is 14, so a(252) = 14.
From Gus Wiseman, Feb 28 2021: (Start)
The strictly inferior divisors of selected n:
n = 1  2  6  12  20  30  42  56  72  90  110  132  156  182  210  240
    -----------------------------------------------------------------
    {} 1  1  1   1   1   1   1   1   1   1    1    1    1    1    1
          2  2   2   2   2   2   2   2   2    2    2    2    2    2
             3   4   3   3   4   3   3   5    3    3    7    3    3
                     5   6   7   4   5   10   4    4    13   5    4
                                 6   6        6    6         6    5
                                 8   9        11   12        7    6
                                                             10   8
                                                             14   10
                                                                  12
                                                                  15
(End)

Maple

with(numtheory):
a:= n-> max(select(d-> is(d=1 or d<sqrt(n)), divisors(n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 29 2018

Mathematica

Table[Part[Divisors[w], Floor[DivisorSigma[0, w]/2]], {w, 1, 256}]
Table[If[n==1, 1, Max[Select[Divisors[n], #<n/#&]]], {n, 100}] (* Gus Wiseman, Feb 28 2021 *)

Prog

(PARI) for (n=2, 1000, d=divisors(n); write("b060775.txt", n, " ", d[length(d)\2])) \\ Harry J. Smith, Jul 11 2009
(PARI) A060775(n)=if(n>1, divisors(n)[numdiv(n)\2], 1) \\ M. F. Hasler, Sep 21 2011

Crossrefs

Cf. A033677, A000196, A000005, A000142, A027423, A055226, A060776, A060777, A002378.

The weakly inferior version is A033676.

Positions of first appearances are A180291.

These are the row-maxima of A341674.

A038548 counts superior (or inferior) divisors.

A056924 counts strictly superior (or strictly inferior) divisors.

A070039 adds up strictly inferior divisors.

A207375 lists central divisors.

A333805 counts strictly inferior odd divisors.

A333806 counts strictly inferior prime divisors.

A341596 counts strictly inferior squarefree divisors.

A341677 counts strictly inferior prime-power divisors.

- Inferior: A063962, A066839, A069288, A161906, A217581, A333749, A333750.

- Superior: A051283, A059172, A063538, A063539, A070038, A161908, A341591.

- Strictly Superior: A048098, A064052, A140271, A238535, A341642, A341673.

Cf. A000203, A001248, A006530, A020639, A112798, A161901.

Sequence in context: A366510 A366522 A345934 * A355368 A175494 A055399

Adjacent sequences: A060772 A060773 A060774 * A060776 A060777 A060778

Keyword

nonn,look

Author

Labos Elemer, Apr 26 2001

Extensions

a(1) = 1 added (to preserve the relation a(n) | n) by Franklin T. Adams-Watters, Jan 27 2018

Edited by M. F. Hasler, Apr 03 2019

Name changed by Gus Wiseman, Feb 28 2021 (was: Lower central (median) divisor of n, with a(1) = 1.)

Status

approved