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 A032742 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). 238
 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, 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, 19, 29, 1, 30, 1, 31, 21, 32, 13, 33, 1, 34, 23, 35, 1, 36, 1, 37, 25, 38, 11, 39, 1, 40 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS It seems that a(n) = Max_{j=n+1..2n-1} gcd(n,j). - Labos Elemer, May 22 2002 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 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 LINKS Rémi Eismann, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe) Moosa Nasir, Slopes. Eric Weisstein's World of Mathematics, Proper Divisor. FORMULA a(n) = n / A020639(n). Other identities and observations: 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 a(n) = A130064(n) / A006530(n). - Reinhard Zumkeller, May 05 2007 a(m)*a(n) < a(m*n) for m and n > 1. - Reinhard Zumkeller, Apr 11 2008 a(m*n) = max(m*a(n), n*a(m)). - Robert Israel, Dec 18 2014 From Antti Karttunen, Mar 31 2018: (Start) a(n) = n - A060681(n). 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. a(n) = A250246(A302042(A250245(n))) = A302026(A302032(A302025(n))). For all n >= 1, A276085(a(A276086(n))) = A276151(n). (End) 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 MAPLE 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 1, seq(n/min(numtheory:-factorset(n)), n=2..1000); # Robert Israel, Dec 18 2014 MATHEMATICA f[n_] := If[n == 1, 1, Divisors[n][[-2]]]; Table[f[n], {n, 100}] (* Vladimir Joseph Stephan Orlovsky, Mar 03 2010 *) Join[{1}, Divisors[#][[-2]]&/@Range[2, 80]] (* Harvey P. Dale, Nov 29 2011 *) a[n_] := n/FactorInteger[n][[1, 1]]; Array[a, 100] (* Amiram Eldar, Nov 26 2020 *) Table[Which[n==1, 1, PrimeQ[n], 1, True, Divisors[n][[-2]]], {n, 80}] (* Harvey P. Dale, Feb 02 2022 *) PROG (PARI) a(n)=if(n==1, 1, n/factor(n)[1, 1]) \\ Charles R Greathouse IV, Jun 15 2011 (Haskell) a032742 n = n `div` a020639 n -- Reinhard Zumkeller, Oct 03 2012 (Scheme) (define (A032742 n) (/ n (A020639 n))) ;; Antti Karttunen, Dec 18 2014 (Python) from sympy import factorint def a(n): return 1 if n == 1 else n//min(factorint(n)) print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Jun 21 2022 CROSSREFS 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. 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). Sequence in context: A325641 A325563 A159353 * A060654 A291329 A291328 Adjacent sequences: A032739 A032740 A032741 * A032743 A032744 A032745 KEYWORD nonn,easy,nice AUTHOR Patrick De Geest, May 15 1998 EXTENSIONS Definition clarified by N. J. A. Sloane, Dec 26 2022 STATUS approved

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Last modified September 28 02:54 EDT 2023. Contains 365714 sequences. (Running on oeis4.)