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A032742 a(1) = 1; for n > 1, a(n) = largest proper divisor of n. 108
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

Seems equal to a(n)=Max{GCD[n,j]; j=n+1,..,2n-1} - Labos Elemer, May 22 2002. This is correct: No integer in 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

LINKS

T. D. Noe and RĂ©mi Eismann, Table of n, a(n) for n=1..10000 (first 1000 terms from T. D. Noe)

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

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 *)

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

CROSSREFS

Equals n/A020639(n).

Cf. A032741, A052126, A068319, A063928.

Sequence in context: A247892 A079880 A159353 * A060654 A280497 A280495

Adjacent sequences:  A032739 A032740 A032741 * A032743 A032744 A032745

KEYWORD

nonn,easy,nice

AUTHOR

Patrick De Geest, May 15 1998

STATUS

approved

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Last modified March 29 18:39 EDT 2017. Contains 284273 sequences.