A226898 -id:A226898 - cp's OEIS frontend

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

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", ")))

A226900 Record values of Hooley's Delta function A226898.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 27, 28, 29, 31, 32, 33, 34, 39, 44, 48, 51, 55, 60, 62, 66, 69, 77, 80, 83, 88, 91, 92, 98, 100, 106, 107, 111, 118, 121, 122, 124, 137, 138, 142, 150, 156

Offset: 1

Charles R Greathouse IV, Jun 21 2013

nonn

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, A226899, A002183.

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]r,r=t;print1(t", ")))

a(38)-a(54) from Charles R Greathouse IV, Jun 24 2013

a(55)-a(56) from Charles R Greathouse IV, Jul 01 2013

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

Original entry on oeis.org

0, 1, 2, 12, 24, 120, 180, 360, 720, 1260, 2880, 3780, 2520, 9240, 5040, 10080, 18480, 15120, 27720, 30240, 50400, 55440, 65520, 83160, 151200, 110880, 240240, 166320, 221760, 277200, 388080, 332640, 360360, 554400, 665280, 786240, 997920, 831600, 942480, 720720, 1995840

Offset: 0

Robert G. Wilson v, Jul 20 2019

nonn

Cf. A226898, A226899, A226900, A226901.

f[n_] := Block[{ds = Divisors@n, k = 1, mx = 0}, len = Length@ds; While[k < len, a = Length@Select[Take[ds, {k, len}], ds[[k]]*E > # &]; If[a > mx, mx = a]; k++]; mx]; f[1] = 1; t[_] := 0; k = 1; While[k < 10^5, a = f@k; If[ t[a] == 0, t[a] = k]; k++]; t@# & /@ Range[0, 23]

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.

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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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A226900 Record values of Hooley's Delta function A226898.

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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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A226901 Partial sums of Hooley's Delta function.

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