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A033676 Largest divisor of n <= sqrt(n). 116
1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 4, 1, 3, 1, 4, 3, 2, 1, 4, 5, 2, 3, 4, 1, 5, 1, 4, 3, 2, 5, 6, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 6, 7, 5, 3, 4, 1, 6, 5, 7, 3, 2, 1, 6, 1, 2, 7, 8, 5, 6, 1, 4, 3, 7, 1, 8, 1, 2, 5, 4, 7, 6, 1, 8, 9, 2, 1, 7, 5, 2, 3 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

a(n) = sqrt(n) is a new record if and only if n is a square. - Zak Seidov, Jul 17 2009

a(n) = A060775(n) unless n is a square, when a(n) = A033677(n) = sqrt(n) is strictly larger than A060775(n). It would be nice to have an efficient algorithm to calculate these terms when n has a large number of divisors, as for example in A060776, A060777 and related problems such as A182987. - M. F. Hasler, Sep 20 2011

a(n) = 1 when n = 1 or n is prime. - Alonso del Arte, Nov 25 2012

a(n) is the smallest central divisor of n. Column 1 of A207375. - Omar E. Pol, Feb 26 2019

REFERENCES

G. Tenenbaum, pp. 268 ff, in: 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

A033677(n) * a(n) = n.

a(n) = A161906(n,A038548(n)). - Reinhard Zumkeller, Mar 08 2013

a(n) = A162348(2n-1). - Daniel Forgues, Sep 29 2014

MAPLE

A033676 := proc(n) local a, d; a := 0 ; for d in numtheory[divisors](n) do if d^2 <= n then a := max(a, d) ; end if; end do: a; end proc: # R. J. Mathar, Aug 09 2009

MATHEMATICA

largestDivisorLEQR[n_Integer] := Module[{dvs = Divisors[n]}, dvs[[Ceiling[Length@dvs/2]]]]; largestDivisorLEQR /@ Range[100] (* Borislav Stanimirov, Mar 28 2010 *)

Table[Last[Select[Divisors[n], #<=Sqrt[n]&]], {n, 100}] (* Harvey P. Dale, Mar 17 2017 *)

PROG

(PARI) A033676(n) = {local(d); if(n<2, 1, d=divisors(n); d[(length(d)+1)\2])} \\ Michael B. Porter, Jan 30 2010

(Haskell)

a033676 n = last $ takeWhile (<= a000196 n) $ a027750_row n

-- Reinhard Zumkeller, Jun 04 2012

(Python)

from sympy import divisors

def A033676(n):

    d = divisors(n)

    return d[(len(d)-1)//2]  # Chai Wah Wu, Apr 05 2021

CROSSREFS

Cf. A033677, A000196, A027750, A056737, A219695, A282668 (positions of primes)

From Omar E. Pol, Jul 05 2009: (Start)

Sequence of corresponding indices:

.. 1 ..... A008578 (1 together with the prime numbers)

.. 2 ..... A161344

.. 3 ..... A161345

.. 4 ..... A161424

.. 5 ..... A161835

.. 6 ..... A162526

.. 7 ..... A162527

.. 8 ..... A162528

.. 9 ..... A162529

. 10 ..... A162530

. 11 ..... A162531

. 12 ..... A162532 (End)

Cf. A207375.

Sequence in context: A217581 A124044 A059981 * A095165 A046805 A034880

Adjacent sequences:  A033673 A033674 A033675 * A033677 A033678 A033679

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 5 23:41 EDT 2021. Contains 343579 sequences. (Running on oeis4.)