OFFSET
1,3
COMMENTS
Is a(n) the least m > 0 such that n - m divides n! + m? - Clark Kimberling, Jul 28 2012
Is a(n) the least m > 0 such that L(n-m) divides L(n+m), where L = A000032 (Lucas numbers)? - Clark Kimberling, Jul 30 2012
Records give A006093. - Omar E. Pol, Oct 26 2013
Divide n by its smallest prime factor p, then multiply with (p-1), with a(1) = 0 by convention. Compare also to A366387. - Antti Karttunen, Oct 23 2023
a(n) is also the smallest LCM of positive integers x and y where x + y = n. - Felix Huber, Aug 28 2024
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
A. Balog, P. Erdős and G. Tenenbaum, On Arithmetic Functions Involving Consecutive Divisors, In: Analytical Number Theory, pp. 77-90, Birkhäuser, Basel, 1990.
FORMULA
a(n) = n - n/A020639(n).
a(n) = n - A032742(n). - Omar E. Pol, Aug 31 2011
a(2n) = n, a(3*(2n+1)) = 2*(2n+1) = 4n + 2. - Antti Karttunen, Oct 23 2023
EXAMPLE
For n = 35, divisors are {1, 5, 7, 35}; differences are {4, 2, 28}; a(35) = largest difference = 28 = 35 - 35/5.
MAPLE
read("transforms") :
A060681 := proc(n)
if n = 1 then
0 ;
else
sort(convert(numtheory[divisors](n), list)) ;
DIFF(%) ;
max(op(%)) ;
end if;
end proc:
seq(A060681(n), n=1..60) ; # R. J. Mathar, May 23 2018
# second Maple program:
A060681:=n->if(n=1, 0, min(map(x->ilcm(x, n-x), [$1..1/2*n]))); seq(A060681(n), n=1..74); # Felix Huber, Aug 28 2024
MATHEMATICA
a[n_ ] := n - n/FactorInteger[n][[1, 1]]
Array[Max[Differences[Divisors[#]]] &, 80, 2] (* Harvey P. Dale, Oct 26 2013 *)
PROG
(Haskell)
a060681 n = div n p * (p - 1) where p = a020639 n
-- Reinhard Zumkeller, Apr 06 2015
(PARI) diff(v)=vector(#v-1, i, v[i+1]-v[i])
a(n)=vecmax(diff(divisors(n))) \\ Charles R Greathouse IV, Sep 02 2015
(PARI) a(n) = if (n==1, 0, n - n/factor(n)[1, 1]); \\ Michel Marcus, Oct 24 2015
(PARI) first(n) = n = max(n, 1); my(res = vector(n)); res[1] = 0; forprime(p = 2, n, for(i = 1, n \ p, if(res[p * i] == 0, res[p * i] = i*(p-1)))); res \\ David A. Corneth, Jan 08 2019
(Python)
from sympy import primefactors
def A060681(n): return n-n//min(primefactors(n), default=1) # Chai Wah Wu, Jun 21 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Labos Elemer, Apr 19 2001
EXTENSIONS
Edited by Dean Hickerson, Jan 22 2002
a(1)=0 added by N. J. A. Sloane, Oct 01 2015 at the suggestion of Antti Karttunen
STATUS
approved