A033677 Smallest divisor of n >= sqrt(n).

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

a(p) = p for p a prime or 1, these are also the record high points in this sequence. - Charles Kusniec, Aug 26 2022

a(n^4+n^2+1) = n^2+n+1 (see A033676). - Jianing Song, Oct 23 2022

Examples

			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)
		

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

Crossrefs

Cf. A027424, A056737, A060775, A219695.

The lower central divisor is A033676.

The strictly superior case is A140271.

Leftmost 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.

Programs

Haskell

a033677 n = head $
   dropWhile ((< n) . (^ 2)) [d | d <- [1..n], mod n d == 0]
-- Reinhard Zumkeller, Oct 20 2011

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 *)

PARI

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

Python

from sympy import divisors
def A033677(n):
    d = divisors(n)
    return d[len(d)//2]  # Chai Wah Wu, Apr 05 2021

Formula

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

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