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A226901 Partial sums of Hooley's Delta function. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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

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 A103877

Adjacent sequences:  A226898 A226899 A226900 * A226902 A226903 A226904

KEYWORD

nonn

AUTHOR

Charles R Greathouse IV, Jun 21 2013

STATUS

approved

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Last modified June 19 16:48 EDT 2019. Contains 324222 sequences. (Running on oeis4.)