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A033676 Largest divisor of n <= sqrt(n). 61
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

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

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)

Sequence in context: A217581 A124044 A059981 * A095165 A046805 A034880

Adjacent sequences:  A033673 A033674 A033675 * A033677 A033678 A033679

KEYWORD

nonn,easy,nice,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified March 23 01:23 EDT 2017. Contains 283901 sequences.