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A226898
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Hooley's Delta function: maximum number of divisors of n in [u, eu] for all u. (Here e is Euler's number 2.718... = A001113.)
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5
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1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 2, 3, 1, 2, 1, 4, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 1, 2, 1, 2, 2, 3, 1, 2, 1, 4, 1, 2, 2, 2, 2, 2, 1, 2, 1, 3, 1, 4, 1, 2, 2, 2, 2, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 4, 2, 2
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OFFSET
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1,2
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COMMENTS
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This function measures the tendency of divisors of a number to cluster.
Tenenbaum (1985) proves that a(1) + ... + a(n) < n exp(c sqrt(log log n log log log n)) for some constant c > 0 and all n > 16. In particular, the average order of a(n) is O((log n)^k) for any k > 0.
Maier & Tenenbaum show that (log log n)^(g + o(1)) < a(n) < (log log n)^(log 2 + o(1)) for almost all n, with g = log 2/log((1-1/log 27)/(1-1/log 3)) = 0.338....
For generalizations, see de la Bretèche & Tenenbaum, Brüdern, Hall & Tenenbaum, and Caballero.
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REFERENCES
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R. R. Hall and G. Tenenbaum, On the average and normal orders of Hooley's ∆-function, J. London Math. Soc. (2), Vol. 25, No. 3 (1982), pp. 392-406.
R. R. Hall and G. Tenenbaum, Divisors. Cambridge Tracts in Mathematics, 90. Cambridge University Press, Cambridge, 1988.
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LINKS
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FORMULA
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a(mn) <= d(m)a(n) where d(n) is A000005.
The average order is between log log x and (log log x)^(11/4); the lower bound is due to Hall & Tenenbaum (1988) and the upper bound to Koukoulopoulos & Tao. - Charles R Greathouse IV, Jun 26 2023
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EXAMPLE
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The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. For u = 3, {3, 4, 6, 8} are in [3, 3e] = [3, 8.15...] and thus a(24) = 4.
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MAPLE
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with(numtheory):
a:= n-> (l-> max(seq(nops(select(x-> is(x<=exp(1)*l[i]), l))-i+1,
i=1..nops(l))))(sort([divisors(n)[]])):
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MATHEMATICA
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a[n_] := Module[{d = Divisors[n], m = 1}, For[i = 1, i < Length[d], i++, t = E*d[[i]]; m = Max[ Sum[ Boole[d[[j]] < t], {j, i, Length[d]}], m]]; m]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 08 2013, after Pari *)
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PROG
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(PARI) a(n)=my(d=divisors(n), m=1); for(i=1, #d-1, my(t=exp(1)*d[i]); m=max(sum(j=i, #d, d[j]<t), m)); m
(PARI) a(n)=my(d=divisors(n), r, t); for(i=1, #d\2, t=setsearch(d, d[i]*exp(1)\1, 1); t=if(t, t-i, setsearch(d, d[i]*exp(1)\1)+1-i); if(t>r, r=t)); r \\ Charles R Greathouse IV, Mar 01 2018
(Haskell)
a226898 = maximum . map length .
map (\ds@(d:_) -> takeWhile (<= e' d) ds) . init . tails . a027750_row
where e' = floor . (* e) . fromIntegral; e = exp 1
(Python)
from sympy import divisors, exp
def a(n):
d = divisors(n)
m = 1
for i in range(len(d) - 1):
t = exp(1)*d[i]
m = max(sum(1 for j in range(i, len(d)) if d[j]<t), m)
return m
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CROSSREFS
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KEYWORD
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nonn,nice,core
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AUTHOR
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STATUS
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approved
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