A226901 - OEIS

%I #13 Jun 27 2023 08:09:22

%S 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,

%T 45,46,48,49,52,53,55,56,58,60,63,64,66,67,71,72,75,76,78,80,82,83,87,

%U 88,90,91,93,94,96,98,101,102,104,105,109,110,112,114,116,118,120,121

%N Partial sums of Hooley's Delta function.

%C 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.

%D 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 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.

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

%H Charles R Greathouse IV, <a href="/A226901/b226901.txt">Table of n, a(n) for n = 1..10000</a>

%H Dimitris Koukoulopoulos and Terence Tao, <a href="https://arxiv.org/abs/2306.08615">A note on the mean value of the Erdős-Hooley Delta function</a>, arXiv preprint (2023). arXiv:2306.08615 [math.NT]

%F 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.

%p with(numtheory):

%p b:= n-> (l-> max(seq(nops(select(x-> is(x<=exp(1)*l[i]), l))-i+1,

%p         i=1..nops(l))))(sort([divisors(n)[]])):

%p a:= proc(n) a(n):= b(n) +`if`(n=1, 0, a(n-1)) end:

%p seq(a(n), n=1..100);  # _Alois P. Heinz_, Jun 21 2013

%t 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];

%t A226901 = Array[delta, 100] // Accumulate (* _Jean-François Alcover_, Mar 24 2017, translated from PARI *)

%o (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]<t), m)); m

%o s=0; vector(100,n,s+=Delta(n))

%Y Partial sums of A226898.

%Y Cf. A226899, A226900.

%K nonn

%O 1,2

%A _Charles R Greathouse IV_, Jun 21 2013