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

%I A033677 #94 Oct 23 2022 16:11:39
%S A033677 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,
%T A033677 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,
%U A033677 19,29,59,10,61,31,9,8,13,11,67,17,23,10,71,9,73,37,15,19,11,13,79,10
%N A033677 Smallest divisor of n >= sqrt(n).
%C A033677 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.
%C A033677 a(n) is the largest central divisor of n. Right border of A207375. - _Omar E. Pol_, Feb 26 2019
%C A033677 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
%C A033677 a(p) = p for p a prime or 1, these are also the record high points in this sequence. - _Charles Kusniec_, Aug 26 2022
%C A033677 a(n^4+n^2+1) = n^2+n+1 (see A033676). - _Jianing Song_, Oct 23 2022
%D A033677 G. Tenenbaum, pp. 268ff of R. L. Graham et al., eds., Mathematics of Paul Erdős I.
%H A033677 T. D. Noe, <a href="/A033677/b033677.txt">Table of n, a(n) for n = 1..10000</a>
%F A033677 a(n) = n/A033676(n).
%F A033677 a(n) = A162348(2n). - _Daniel Forgues_, Sep 29 2014
%e A033677 From _Gus Wiseman_, Feb 19 2021: (Start)
%e A033677 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.
%e A033677 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.
%e A033677 (End)
%p A033677 A033677 := proc(n)
%p A033677     n/A033676(n) ;
%p A033677 end proc:
%t A033677 Table[Select[Divisors[n], # >= Sqrt[n] &, 1] // First, {n, 80}]  (* _Jean-François Alcover_, Apr 01 2011 *)
%o A033677 (PARI) A033677(n) = {local(d); d=divisors(n); d[length(d)\2+1]} \\ _Michael B. Porter_, Feb 26 2010
%o A033677 (Haskell)
%o A033677 a033677 n = head $
%o A033677    dropWhile ((< n) . (^ 2)) [d | d <- [1..n], mod n d == 0]
%o A033677 -- _Reinhard Zumkeller_, Oct 20 2011
%o A033677 (Python)
%o A033677 from sympy import divisors
%o A033677 def A033677(n):
%o A033677     d = divisors(n)
%o A033677     return d[len(d)//2]  # _Chai Wah Wu_, Apr 05 2021
%Y A033677 Cf. A027424, A056737, A060775, A219695.
%Y A033677 The lower central divisor is A033676.
%Y A033677 The strictly superior case is A140271.
%Y A033677 Leftmost column of A161908 (superior divisors).
%Y A033677 Rightmost column of A207375 (central divisors).
%Y A033677 A038548 counts superior (or inferior) divisors.
%Y A033677 A056924 counts strictly superior (or strictly inferior) divisors.
%Y A033677 A063538/A063539 list numbers with/without a superior prime divisor.
%Y A033677 A070038 adds up superior divisors.
%Y A033677 A341676 selects the unique superior prime divisor.
%Y A033677 - Inferior: A063962, A066839, A069288, A161906, A217581, A333749, A333750.
%Y A033677 - Superior: A059172, A116882, A116883, A341591, A341592, A341593, A341675.
%Y A033677 - Strictly Inferior: A070039, A333805, A333806, A341596, A341674, A341677.
%Y A033677 - Strictly Superior: A048098, A064052, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646, A341673.
%Y A033677 Cf. A000005, A000203, A001248, A001221, A001222, A020639, A051283.
%K A033677 nonn,easy,nice
%O A033677 1,2
%A A033677 _N. J. A. Sloane_