A226900 -id:A226900 - cp's OEIS frontend

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.)

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

Offset: 1

Charles R Greathouse IV, Jun 21 2013

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.

			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.

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.

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.
Index entries for "core" sequences

Partial sums are A226901. Cf. A226899, A226900, A027750, A022843.

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

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

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 *)

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]

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

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]Indranil Ghosh, Jul 19 2017

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

A226899 Numbers n for which Delta(m) < Delta(n) for all m < n, where Delta is Hooley's Delta function A226898.

Original entry on oeis.org

1, 2, 12, 24, 120, 180, 360, 720, 1260, 2520, 5040, 10080, 15120, 27720, 30240, 50400, 55440, 65520, 83160, 110880, 166320, 221760, 277200, 332640, 360360, 554400, 665280, 720720, 1441440, 2162160, 3603600, 4324320, 7207200, 8648640, 10810800, 14414400, 21621600

Offset: 1

Charles R Greathouse IV, Jun 21 2013

nonn

Not a subsequence of A025487: a(18) = 65520 = 2^4 * 3^2 * 5 * 7 * 13, a(46) = 136936800 = 2^5 * 3^2 * 5^2 * 7 * 11 * 13 * 19, and a(50) = 273873600 = 2^6 * 3^2 * 5^2 * 7 * 11 * 13 * 19. - Charles R Greathouse IV, Mar 12 2018

Charles R Greathouse IV, Table of n, a(n) for n = 1..64
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.

Cf. A226898, A226900, A002182.

Delta(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
r=0; for(n=1,1e9, t=Delta(n); if(t>r, r=t; print1(n", ")))

A226901 Partial sums of Hooley's Delta function.

Original entry on oeis.org

1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 19, 20, 22, 24, 26, 27, 29, 30, 33, 35, 37, 38, 42, 43, 45, 46, 48, 49, 52, 53, 55, 56, 58, 60, 63, 64, 66, 67, 71, 72, 75, 76, 78, 80, 82, 83, 87, 88, 90, 91, 93, 94, 96, 98, 101, 102, 104, 105, 109, 110, 112, 114, 116, 118, 120, 121

Offset: 1

Charles R Greathouse IV, Jun 21 2013

nonn

Tenenbaum (1985) proves that a(n) < n exp(c sqrt(log log n log log log n)) for some constant c > 0 and all n > 16. Numerically, c appears to be close to 0.5 or 0.55.

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.
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.
Gérald Tenenbaum, Sur la concentration moyenne des diviseurs, Commentarii Mathematici Helvetici 60:1 (1985), pp. 411-428.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
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]

Partial sums of A226898.

Cf. A226899, A226900.

with(numtheory):
b:= n-> (l-> max(seq(nops(select(x-> is(x<=exp(1)*l[i]), l))-i+1,
        i=1..nops(l))))(sort([divisors(n)[]])):
a:= proc(n) a(n):= b(n) +`if`(n=1, 0, a(n-1)) end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jun 21 2013

delta[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];
A226901 = Array[delta, 100] // Accumulate (* Jean-François Alcover, Mar 24 2017, translated from PARI *)

Delta(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]

n log log n << a(n) << n (log log n)^(11/4); the lower bound is due to Hall & Tenenbaum (1988) and the upper bound to Koukoulopoulos & Tao.

cp's OEIS Frontend

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.)

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A226899 Numbers n for which Delta(m) < Delta(n) for all m < n, where Delta is Hooley's Delta function A226898.

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A226901 Partial sums of Hooley's Delta function.

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A309278 Index of first occurrence of n in the Erdös-Hooley Delta function A226898, with a(0)=0.

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