%I #77 Oct 23 2022 16:11:45
%S 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,
%T 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,
%U 3,7,1,8,1,2,5,4,7,6,1,8,9,2,1,7,5,2,3
%N Largest divisor of n <= sqrt(n).
%C a(n) = sqrt(n) is a new record if and only if n is a square. - _Zak Seidov_, Jul 17 2009
%C 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
%C a(n) = 1 when n = 1 or n is prime. - _Alonso del Arte_, Nov 25 2012
%C a(n) is the smallest central divisor of n. Column 1 of A207375. - _Omar E. Pol_, Feb 26 2019
%C a(n^4+n^2+1) = n^2-n+1: suppose that n^2-n+k divides n^4+n^2+1 = (n^2-n+k)*(n^2+n-k+2) - (k-1)*(2*n+1-k) for 2 <= k <= 2*n, then (k-1)*(2*n+1-k) >= n^2-n+k, or n^2 - (2*k-1)*n + (k^2-k+1) = (n-k+1/2)^2 + 3/4 < 0, which is impossible. Hence the next smallest divisor of n^4+n^2+1 than n^2-n+1 is at least n^2-n+(2*n+1) = n^2+n+1 > sqrt(n^4+n^2+1). - _Jianing Song_, Oct 23 2022
%D G. Tenenbaum, pp. 268 ff, in: R. L. Graham et al., eds., Mathematics of Paul Erdős I.
%H T. D. Noe, <a href="/A033676/b033676.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = n / A033677(n).
%F a(n) = A161906(n,A038548(n)). - _Reinhard Zumkeller_, Mar 08 2013
%F a(n) = A162348(2n-1). - _Daniel Forgues_, Sep 29 2014
%p 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
%t largestDivisorLEQR[n_Integer] := Module[{dvs = Divisors[n]}, dvs[[Ceiling[Length@dvs/2]]]]; largestDivisorLEQR /@ Range[100] (* _Borislav Stanimirov_, Mar 28 2010 *)
%t Table[Last[Select[Divisors[n],#<=Sqrt[n]&]],{n,100}] (* _Harvey P. Dale_, Mar 17 2017 *)
%o (PARI) A033676(n) = {local(d);if(n<2,1,d=divisors(n);d[(length(d)+1)\2])} \\ _Michael B. Porter_, Jan 30 2010
%o (Haskell)
%o a033676 n = last $ takeWhile (<= a000196 n) $ a027750_row n
%o -- _Reinhard Zumkeller_, Jun 04 2012
%o (Python)
%o from sympy import divisors
%o def A033676(n):
%o d = divisors(n)
%o return d[(len(d)-1)//2] # _Chai Wah Wu_, Apr 05 2021
%Y Cf. A033677 (n/a(n)), A000196 (sqrt), A027750 (list of divisors), A056737 (n/a(n) - a(n)), A219695 (half of this for odd numbers), A207375 (list the central divisor(s)).
%Y The strictly inferior case is A060775. Cf. also A140271.
%Y Indices of given values: A008578 (1 and the prime numbers: a(n) = 1), A161344 (a(n) = 2), A161345 (a(n) = 3), A161424 (4), A161835 (5), A162526 (6), A162527 (7), A162528 (8), A162529 (9), A162530 (10), A162531 (11), A162532 (12), A282668 (indices of primes).
%K nonn,easy,nice
%O 1,4
%A _N. J. A. Sloane_