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A226901 Partial sums of Hooley's Delta function.

%I A226901 #13 Jun 27 2023 08:09:22
%S A226901 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 A226901 45,46,48,49,52,53,55,56,58,60,63,64,66,67,71,72,75,76,78,80,82,83,87,
%U A226901 88,90,91,93,94,96,98,101,102,104,105,109,110,112,114,116,118,120,121
%N A226901 Partial sums of Hooley's Delta function.
%C A226901 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 A226901 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 A226901 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 A226901 Gérald Tenenbaum, Sur la concentration moyenne des diviseurs, Commentarii Mathematici Helvetici 60:1 (1985), pp. 411-428.
%H A226901 Charles R Greathouse IV, <a href="/A226901/b226901.txt">Table of n, a(n) for n = 1..10000</a>
%H A226901 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 A226901 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 A226901 with(numtheory):
%p A226901 b:= n-> (l-> max(seq(nops(select(x-> is(x<=exp(1)*l[i]), l))-i+1,
%p A226901         i=1..nops(l))))(sort([divisors(n)[]])):
%p A226901 a:= proc(n) a(n):= b(n) +`if`(n=1, 0, a(n-1)) end:
%p A226901 seq(a(n), n=1..100);  # _Alois P. Heinz_, Jun 21 2013
%t A226901 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 A226901 = Array[delta, 100] // Accumulate (* _Jean-François Alcover_, Mar 24 2017, translated from PARI *)
%o A226901 (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 A226901 s=0; vector(100,n,s+=Delta(n))
%Y A226901 Partial sums of A226898.
%Y A226901 Cf. A226899, A226900.
%K A226901 nonn
%O A226901 1,2
%A A226901 _Charles R Greathouse IV_, Jun 21 2013