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A033676
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Largest divisor of n <= sqrt(n).
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116
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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
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OFFSET
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1,4
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COMMENTS
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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
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REFERENCES
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G. Tenenbaum, pp. 268 ff, in: R. L. Graham et al., eds., Mathematics of Paul Erdős I.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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