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A033677 Smallest divisor of n >= sqrt(n). 82
1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 4, 13, 7, 5, 4, 17, 6, 19, 5, 7, 11, 23, 6, 5, 13, 9, 7, 29, 6, 31, 8, 11, 17, 7, 6, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 8, 7, 10, 17, 13, 53, 9, 11, 8, 19, 29, 59, 10, 61, 31, 9, 8, 13, 11, 67, 17, 23, 10, 71, 9, 73, 37, 15, 19, 11, 13, 79, 10 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

a(n) is the smallest k such that n appears in the k X k multiplication table and A027424(k) is the number of n with a(n) <= k.

a(n) is the largest central divisor of n. Right border of A207375. - Omar E. Pol, Feb 26 2019

If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence selects the smallest superior divisor of n. - Gus Wiseman, Feb 19 2021

REFERENCES

G. Tenenbaum, pp. 268ff of R. L. Graham et al., eds., Mathematics of Paul Erdős I.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

FORMULA

a(n) = n/A033676(n).

a(n) = A162348(2n). - Daniel Forgues, Sep 29 2014

EXAMPLE

From Gus Wiseman, Feb 19 2021: (Start)

The divisors of 36 are {1,2,3,4,6,9,12,18,36}. Of these {1,2,3,4,6} are inferior and {6,9,12,18,36} are superior, so a(36) = 6.

The divisors of 40 are {1,2,4,5,8,10,20,40}. Of these {1,2,4,5} are inferior and {8,10,20,40} are superior, so a(40) = 8.

(End)

MAPLE

A033677 := proc(n)

    n/A033676(n) ;

end proc:

MATHEMATICA

Table[Select[Divisors[n], # >= Sqrt[n] &, 1] // First, {n, 80}]  (* Jean-François Alcover, Apr 01 2011 *)

PROG

(PARI) A033677(n) = {local(d); d=divisors(n); d[length(d)\2+1]} \\ Michael B. Porter, Feb 26 2010

(Haskell)

a033677 n = head $

   dropWhile ((< n) . (^ 2)) [d | d <- [1..n], mod n d == 0]

-- Reinhard Zumkeller, Oct 20 2011

(Python)

from sympy import divisors

def A033677(n):

    d = divisors(n)

    return d[len(d)//2]  # Chai Wah Wu, Apr 05 2021

CROSSREFS

Cf. A027424, A056737, A060775, A219695.

The lower central divisor is A033676.

The strictly superior case is A140271.

Left-most column of A161908 (superior divisors).

Rightmost column of A207375 (central divisors).

A038548 counts superior (or inferior) divisors.

A056924 counts strictly superior (or strictly inferior) divisors.

A063538/A063539 list numbers with/without a superior prime divisor.

A070038 adds up superior divisors.

A341676 selects the unique superior prime divisor.

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

- Superior: A059172, A116882, A116883, A341591, A341592, A341593, A341675.

- Strictly Inferior: A070039, A333805, A333806, A341596, A341674, A341677.

- Strictly Superior: A048098, A064052, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646, A341673.

Cf. A000005, A000203, A001248, A001221, A001222, A020639, A051283.

Sequence in context: A341679 A072505 A095163 * A116548 A348582 A346701

Adjacent sequences:  A033674 A033675 A033676 * A033678 A033679 A033680

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 17 14:02 EDT 2022. Contains 353746 sequences. (Running on oeis4.)