A226901 Partial sums of Hooley's Delta function.

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

1,2

Comments

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.

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.

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.

Links

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]

Formula

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.

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

Partial sums of A226898.

Cf. A226899, A226900.

Sequence in context: A064717 A109231 A140098 * A286323 A059559 A329847

Adjacent sequences: A226898 A226899 A226900 * A226902 A226903 A226904

Keyword

nonn

Author

Charles R Greathouse IV, Jun 21 2013

Status

approved