# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a226898 Showing 1-1 of 1 %I A226898 #101 Jun 27 2023 08:09:40 %S A226898 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, %T A226898 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, %U A226898 1,3,1,4,1,2,2,2,2,2,1,4,1,2,1,4,1,2,1,2,1,4,2,2 %N A226898 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.) %C A226898 This function measures the tendency of divisors of a number to cluster. %C A226898 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. %C A226898 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.... %C A226898 For generalizations, see de la Bretèche & Tenenbaum, Brüdern, Hall & Tenenbaum, and Caballero. %D A226898 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. %D A226898 R. R. Hall and G. Tenenbaum, Divisors. Cambridge Tracts in Mathematics, 90. Cambridge University Press, Cambridge, 1988. %H A226898 Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 %H A226898 R. de la Bretèche and G. Tenenbaum, Oscillations localisées sur les diviseurs, J. Lond. Math. Soc. 2 85:3 (2012), pp. 669-693. %H A226898 Régis de la Bretèche and Gérald Tenenbaum, Two upper bounds for the Erdős--Hooley Delta-function, arXiv preprint (2022). arXiv:2210.13897 [math.NT] %H A226898 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. %H A226898 Paul Erdős, On abundant-like numbers, Canad. Math. Bull. 17 (1974), pp. 599-602. %H A226898 Paul Erdős and Jean-Louis Nicolas, Méthodes probabilistes et combinatoires en théorie des nombres, Bulletin des Sciences Mathématiques 2 (1976), pp. 301-320. %H A226898 P. Erdős and J.-L. Nicolas, Répartition des nombres superabondants, Bull. Soc. Math. France 103 (1975), pp. 65-90. %H A226898 R. R. Hall and G. Tenenbaum, The average orders of Hooley's Δ_r-functions, Mathematika 31:1 (1984), pp. 98-109. %H A226898 R. R. Hall and G. Tenenbaum, The average orders of Hooley's Δ_r-functions, II, Compositio Math. 60 (1986), pp. 163-186. %H A226898 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. %H A226898 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] %H A226898 Helmut Maier and Gérald Tenenbaum, On the set of divisors of an integer, Invent. Math. 76 (1984), pp. 121-128. %H A226898 Helmut Maier and Gérald Tenenbaum, On the normal concentration of divisors, J. London Math. Soc. 2 31:3 (1985), pp. 393-400. %H A226898 Helmut Maier and Gérald Tenenbaum, On the normal concentration of divisors. II., Math. Proc. Cambridge Philos. Soc. 147:3 (2009), pp. 513-540. %H A226898 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. %H A226898 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. %H A226898 Gérald Tenenbaum, Sur la concentration moyenne des diviseurs, Commentarii Mathematici Helvetici 60:1 (1985), pp. 411-428. %H A226898 Index entries for "core" sequences %F A226898 a(mn) <= d(m)a(n) where d(n) is A000005. %F A226898 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 %e A226898 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. %p A226898 with(numtheory): %p A226898 a:= n-> (l-> max(seq(nops(select(x-> is(x<=exp(1)*l[i]), l))-i+1, %p A226898 i=1..nops(l))))(sort([divisors(n)[]])): %p A226898 seq(a(n), n=1..100); # _Alois P. Heinz_, Jun 21 2013 %t A226898 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 *) %o A226898 (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]r, r=t)); r \\ _Charles R Greathouse IV_, Mar 01 2018 %o A226898 (Haskell) %o A226898 a226898 = maximum . map length . %o A226898 map (\ds@(d:_) -> takeWhile (<= e' d) ds) . init . tails . a027750_row %o A226898 where e' = floor . (* e) . fromIntegral; e = exp 1 %o A226898 -- _Reinhard Zumkeller_, Jul 06 2013 %o A226898 (Python) %o A226898 from sympy import divisors, exp %o A226898 def a(n): %o A226898 d = divisors(n) %o A226898 m = 1 %o A226898 for i in range(len(d) - 1): %o A226898 t = exp(1)*d[i] %o A226898 m = max(sum(1 for j in range(i, len(d)) if d[j]