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%I #107 Dec 26 2022 15:43:08
%S 1,1,1,2,1,3,1,4,3,5,1,6,1,7,5,8,1,9,1,10,7,11,1,12,5,13,9,14,1,15,1,
%T 16,11,17,7,18,1,19,13,20,1,21,1,22,15,23,1,24,7,25,17,26,1,27,11,28,
%U 19,29,1,30,1,31,21,32,13,33,1,34,23,35,1,36,1,37,25,38,11,39,1,40
%N a(1) = 1; for n > 1, a(n) = largest proper divisor of n (that is, for n>1, maximum divisor d of n in range 1 <= d < n).
%C It seems that a(n) = Max_{j=n+1..2n-1} gcd(n,j). - _Labos Elemer_, May 22 2002
%C This is correct: No integer in the range [n+1, 2n-1] has n as its divisor, but certainly at least one multiple of the largest proper divisor of n will occur there (e.g., if it is n/2, then at n + (n/2)). - _Antti Karttunen_, Dec 18 2014
%C The slopes of the visible lines made by the points in the scatter plot are 1/2, 1/3, 1/5, 1/7, ... (reciprocals of primes). - _Moosa Nasir_, Jun 19 2022
%H Rémi Eismann, <a href="/A032742/b032742.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H Moosa Nasir, <a href="https://raw.githubusercontent.com/TealEgg/Testing/main/Slopes.png">Slopes</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ProperDivisor.html">Proper Divisor</a>.
%F a(n) = n / A020639(n).
%F Other identities and observations:
%F A054576(n) = a(a(n)); A117358(n) = a(a(a(n))) = a(A054576(n)); a(A008578(n)) = 1, a(A002808(n)) > 1. - _Reinhard Zumkeller_, Mar 10 2006
%F a(n) = A130064(n) / A006530(n). - _Reinhard Zumkeller_, May 05 2007
%F a(m)*a(n) < a(m*n) for m and n > 1. - _Reinhard Zumkeller_, Apr 11 2008
%F a(m*n) = max(m*a(n), n*a(m)). - _Robert Israel_, Dec 18 2014
%F From _Antti Karttunen_, Mar 31 2018: (Start)
%F a(n) = n - A060681(n).
%F For n > 1, a(n) = A003961^(r)(A246277(n)), where r = A055396(n)-1 and A003961^(r)(n) stands for shifting the prime factorization of n by r positions towards larger primes.
%F a(n) = A250246(A302042(A250245(n))) = A302026(A302032(A302025(n))).
%F For all n >= 1, A276085(a(A276086(n))) = A276151(n).
%F (End)
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Sum_{k>=1} A005867(k-1)/(prime(k)*A002110(k)) = 0.165049... . - _Amiram Eldar_, Nov 19 2022
%p A032742 :=proc(n) option remember; if n = 1 then 1; else numtheory[divisors](n) minus {n} ; max(op(%)) ; end if; end proc: # _R. J. Mathar_, Jun 13 2011
%p 1, seq(n/min(numtheory:-factorset(n)), n=2..1000); # _Robert Israel_, Dec 18 2014
%t f[n_] := If[n == 1, 1, Divisors[n][[-2]]]; Table[f[n], {n, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 03 2010 *)
%t Join[{1},Divisors[#][[-2]]&/@Range[2,80]] (* _Harvey P. Dale_, Nov 29 2011 *)
%t a[n_] := n/FactorInteger[n][[1, 1]]; Array[a, 100] (* _Amiram Eldar_, Nov 26 2020 *)
%t Table[Which[n==1,1,PrimeQ[n],1,True,Divisors[n][[-2]]],{n,80}] (* _Harvey P. Dale_, Feb 02 2022 *)
%o (PARI) a(n)=if(n==1,1,n/factor(n)[1,1]) \\ _Charles R Greathouse IV_, Jun 15 2011
%o (Haskell)
%o a032742 n = n `div` a020639 n -- _Reinhard Zumkeller_, Oct 03 2012
%o (Scheme) (define (A032742 n) (/ n (A020639 n))) ;; _Antti Karttunen_, Dec 18 2014
%o (Python)
%o from sympy import factorint
%o def a(n): return 1 if n == 1 else n//min(factorint(n))
%o print([a(n) for n in range(1, 81)]) # _Michael S. Branicky_, Jun 21 2022
%Y Cf. A002110, A002808, A005867, A006530, A008578, A020639, A032741, A003961, A052126, A054576, A055396, A060681, A068319, A063928, A130064, A246277, A250245, A250246, A276085, A276086, A276151, A286477, A300236, A302025, A302026, A302032, A302042, A325563, A325567.
%Y Maximal GCD of k positive integers with sum n for k = 2..10: this sequence (k=2,n>=2), A355249 (k=3), A355319 (k=4), A355366 (k=5), A355368 (k=6), A355402 (k=7), A354598 (k=8), A354599 (k=9), A354601 (k=10).
%K nonn,easy,nice
%O 1,4
%A _Patrick De Geest_, May 15 1998
%E Definition clarified by _N. J. A. Sloane_, Dec 26 2022
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