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a(n) is the greatest divisor of n^2 that is less than n.
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%I #12 Jun 25 2018 22:57:30

%S 1,1,2,1,4,1,4,3,5,1,9,1,7,9,8,1,12,1,16,9,11,1,18,5,13,9,16,1,25,1,

%T 16,11,17,25,27,1,19,13,32,1,36,1,22,27,23,1,36,7,25,17,26,1,36,25,49,

%U 19,29,1,50,1,31,49,32,25,44,1,34,23,50,1,64,1,37,45,38,49,52,1,64,27,41

%N a(n) is the greatest divisor of n^2 that is less than n.

%C Smaller of two distinct numbers with minimum sum whose geometric mean is n. E.g., a(12) = 9 as 12^2 = 144 = 1*144 = 2*72 = 3*48 = 4*36 = 6*24 = 8*18 = 9*16, etc. - _Amarnath Murthy_, Feb 15 2003

%H Harry J. Smith, <a href="/A063717/b063717.txt">Table of n, a(n) for n = 2..1000</a>

%e a(45)=27 because set of divisors of 45^2 is {1, 3, 5, 9, 15, 25, 27, 45, 75, 81, 135, 225, 405, 675, 2025} and the greatest element of the set less than 45 is 27.

%p with(numtheory): for n from 2 to 200 do a := divisors(n^2): b := a[(tau(n^2)-1)/2]: printf(`%d,`,b); od:

%t f[n_] := Module[{dn2 = Divisors[n^2]}, Last[Take[dn2, {1, Flatten[Position[dn2, n]][[ 1]] - 1}]]]; Table[f[i], {i, 2, 85}]

%t Table[Select[Divisors[n^2],#<n&][[-1]],{n,2,100}] (* _Harvey P. Dale_, Apr 23 2016 *)

%o (PARI) { for (n=2, 1000, d=divisors(n^2); write("b063717.txt", n, " ", d[length(d)\2]) ) } \\ _Harry J. Smith_, Aug 28 2009

%Y A063649(n) = n + a(n), A063718(n) = n^2/A063717(n), A063428(n) = n - a(n).

%Y Cf. A063718.

%K nonn

%O 2,3

%A _Vladeta Jovovic_, Aug 12 2001