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A060681
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Largest difference between consecutive divisors of n (ordered by size).
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70
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0, 1, 2, 2, 4, 3, 6, 4, 6, 5, 10, 6, 12, 7, 10, 8, 16, 9, 18, 10, 14, 11, 22, 12, 20, 13, 18, 14, 28, 15, 30, 16, 22, 17, 28, 18, 36, 19, 26, 20, 40, 21, 42, 22, 30, 23, 46, 24, 42, 25, 34, 26, 52, 27, 44, 28, 38, 29, 58, 30, 60, 31, 42, 32, 52, 33, 66, 34, 46, 35, 70, 36, 72, 37
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OFFSET
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1,3
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COMMENTS
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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
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
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LINKS
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FORMULA
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EXAMPLE
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For n = 35, divisors are {1, 5, 7, 35}; differences are {4, 2, 28}; a(35) = largest difference = 28 = 35 - 35/5.
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MAPLE
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read("transforms") :
if n = 1 then
0 ;
else
sort(convert(numtheory[divisors](n), list)) ;
DIFF(%) ;
max(op(%)) ;
end if;
end proc:
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MATHEMATICA
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a[n_ ] := n - n/FactorInteger[n][[1, 1]]
Array[Max[Differences[Divisors[#]]] &, 80, 2] (* Harvey P. Dale, Oct 26 2013 *)
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PROG
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(Haskell)
a060681 n = div n p * (p - 1) where p = a020639 n
(PARI) diff(v)=vector(#v-1, i, v[i+1]-v[i])
(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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