OFFSET
1,2
COMMENTS
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.
REFERENCES
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.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
R. de la Bretèche and G. Tenenbaum, Oscillations localisées sur les diviseurs, J. Lond. Math. Soc. 2 85:3 (2012), pp. 669-693.
Régis de la Bretèche and Gérald Tenenbaum, Two upper bounds for the Erdős--Hooley Delta-function, arXiv preprint (2022). arXiv:2210.13897 [math.NT]
Jörg Brüdern, Daniel's twists of Hooley's Delta function, Contributions in Analytic and Algebraic Number Theory, Springer Proceedings in Mathematics 9 (2012), pp 31-82.
Paul Erdős, On abundant-like numbers, Canad. Math. Bull. 17 (1974), pp. 599-602.
Paul Erdős and Jean-Louis Nicolas, Méthodes probabilistes et combinatoires en théorie des nombres, Bulletin des Sciences Mathématiques 2 (1976), pp. 301-320.
P. Erdős and J.-L. Nicolas, Répartition des nombres superabondants, Bull. Soc. Math. France 103 (1975), pp. 65-90.
R. R. Hall and G. Tenenbaum, The average orders of Hooley's Δ_r-functions, Mathematika 31:1 (1984), pp. 98-109.
R. R. Hall and G. Tenenbaum, The average orders of Hooley's Δ_r-functions, II, Compositio Math. 60 (1986), pp. 163-186.
C. Hooley, On a new technique and its applications to the theory of numbers, Proc. London Math. Soc. 3 38:1 (1979), pp. 115-151.
Dimitris Koukoulopoulos and Terence Tao, A note on the mean value of the Erdős-Hooley Delta function, arXiv preprint (2023). arXiv:2306.08615 [math.NT]
Helmut Maier and Gérald Tenenbaum, On the set of divisors of an integer, Invent. Math. 76 (1984), pp. 121-128.
Helmut Maier and Gérald Tenenbaum, On the normal concentration of divisors, J. London Math. Soc. 2 31:3 (1985), pp. 393-400.
Helmut Maier and Gérald Tenenbaum, On the normal concentration of divisors. II., Math. Proc. Cambridge Philos. Soc. 147:3 (2009), pp. 513-540.
J.-L. Nicolas, Méthodes probabilistes et combinatoires en théorie des nombres, Séminaire Delange-Pisot-Poitou. Théorie des nombres, Tome 17 (1975-1976) no. 1, Exposé no. 9, p. 1.
J. M. Rodríguez Caballero, Symmetric Dyck Paths and Hooley's Δ-Function, In: Brlek S., Dolce F., Reutenauer C., Vandomme É. (eds) Combinatorics on Words, WORDS 2017, Lecture Notes in Computer Science, vol 10432.
Gérald Tenenbaum, Sur la concentration moyenne des diviseurs, Commentarii Mathematici Helvetici 60:1 (1985), pp. 411-428.
FORMULA
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
EXAMPLE
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.
MAPLE
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)[]])):
seq(a(n), n=1..100); # Alois P. Heinz, Jun 21 2013
MATHEMATICA
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 *)
PROG
(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
-- Reinhard Zumkeller, Jul 06 2013
(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
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Jul 19 2017
CROSSREFS
KEYWORD
nonn,nice,core
AUTHOR
Charles R Greathouse IV, Jun 21 2013
STATUS
approved