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%I #44 Sep 30 2022 12:50:15
%S 1,1,1,1,1,2,1,2,1,2,1,3,1,2,3,2,1,3,1,4,3,2,1,4,1,2,3,4,1,5,1,4,3,2,
%T 5,4,1,2,3,5,1,6,1,4,5,2,1,6,1,5,3,4,1,6,5,7,3,2,1,6,1,2,7,4,5,6,1,4,
%U 3,7,1,8,1,2,5,4,7,6,1,8,3,2,1,7,5,2,3
%N The greatest divisor d|n such that d < n/d, with a(1) = 1.
%C Also: Largest divisor of n which is less than sqrt(n).
%C If n is not a square, then a(n) = A033676(n), else a(n) is strictly smaller than A033676(n) = sqrt(n) (except for a(1) = 1). - _M. F. Hasler_, Sep 20 2011
%C Record values occur for n = k * (k+1), for which a(n) = k. - _Franklin T. Adams-Watters_, May 01 2015
%C If we define a divisor d|n to be strictly inferior if d < n/d, then strictly inferior divisors are counted by A056924 and listed by A341674. This sequence gives the greatest strictly inferior divisor, which may differ from the lower central divisor A033676. Central divisors are listed by A207375. - _Gus Wiseman_, Feb 28 2021
%H Alois P. Heinz, <a href="/A060775/b060775.txt">Table of n, a(n) for n = 1..10000</a> (terms n = 2..1000 from Harry J. Smith)
%F a(n) = max { d: d|n and d < sqrt(n) or d = 1 }, where "|" means "divides". [Corrected by _M. F. Hasler_, Apr 03 2019]
%e n = 252, D = {1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252}, 18 divisors, the 9th is 14, so a(252) = 14.
%e From _Gus Wiseman_, Feb 28 2021: (Start)
%e The strictly inferior divisors of selected n:
%e n = 1 2 6 12 20 30 42 56 72 90 110 132 156 182 210 240
%e -----------------------------------------------------------------
%e {} 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
%e 2 2 2 2 2 2 2 2 2 2 2 2 2 2
%e 3 4 3 3 4 3 3 5 3 3 7 3 3
%e 5 6 7 4 5 10 4 4 13 5 4
%e 6 6 6 6 6 5
%e 8 9 11 12 7 6
%e 10 8
%e 14 10
%e 12
%e 15
%e (End)
%p with(numtheory):
%p a:= n-> max(select(d-> is(d=1 or d<sqrt(n)), divisors(n))):
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Jan 29 2018
%t Table[Part[Divisors[w], Floor[DivisorSigma[0, w]/2]], {w, 1, 256}]
%t Table[If[n==1,1,Max[Select[Divisors[n],#<n/#&]]],{n,100}] (* _Gus Wiseman_, Feb 28 2021 *)
%o (PARI) for (n=2, 1000, d=divisors(n); write("b060775.txt", n, " ", d[length(d)\2])) \\ _Harry J. Smith_, Jul 11 2009
%o (PARI) A060775(n)=if(n>1,divisors(n)[numdiv(n)\2],1) \\ _M. F. Hasler_, Sep 21 2011
%Y Cf. A033677, A000196, A000005, A000142, A027423, A055226, A060776, A060777, A002378.
%Y The weakly inferior version is A033676.
%Y Positions of first appearances are A180291.
%Y These are the row-maxima of A341674.
%Y A038548 counts superior (or inferior) divisors.
%Y A056924 counts strictly superior (or strictly inferior) divisors.
%Y A070039 adds up strictly inferior divisors.
%Y A207375 lists central divisors.
%Y A333805 counts strictly inferior odd divisors.
%Y A333806 counts strictly inferior prime divisors.
%Y A341596 counts strictly inferior squarefree divisors.
%Y A341677 counts strictly inferior prime-power divisors.
%Y - Inferior: A063962, A066839, A069288, A161906, A217581, A333749, A333750.
%Y - Superior: A051283, A059172, A063538, A063539, A070038, A161908, A341591.
%Y - Strictly Superior: A048098, A064052, A140271, A238535, A341642, A341673.
%Y Cf. A000203, A001248, A006530, A020639, A112798, A161901.
%K nonn,look
%O 1,6
%A _Labos Elemer_, Apr 26 2001
%E a(1) = 1 added (to preserve the relation a(n) | n) by _Franklin T. Adams-Watters_, Jan 27 2018
%E Edited by _M. F. Hasler_, Apr 03 2019
%E Name changed by _Gus Wiseman_, Feb 28 2021 (was: Lower central (median) divisor of n, with a(1) = 1.)
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