A374074 - OEIS

A374074

Odd composite numbers k sorted by k/2^(bigomega(k) - 1).

9, 27, 15, 81, 21, 45, 25, 243, 63, 33, 135, 35, 75, 39, 729, 189, 49, 99, 405, 51, 105, 55, 225, 57, 117, 125, 65, 2187, 69, 567, 147, 297, 1215, 153, 77, 315, 165, 675, 85, 171, 87, 175, 351, 91, 93, 375, 95, 195, 6561, 207, 1701, 441, 111, 891, 3645, 459

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Sorting by k/2^bigomega(k) would give the same sequence.

It appears that this sequence can be used to approximate the imaginary parts of the nontrivial zeta zeros, that is, A002410(n) is roughly equal to 2*Pi*a(n)/2^bigomega(a(n)) - n/2 + sqrt(n)/2.

Calculations show that the relative error approaches 1.0+-0.005 for the first 3800 zeros (z=2000 in Mathematica code). For further zeros, a better approximation may be useful, e.g. 2*Pi*a(n)/2^bigomega(a(n)) - n/2 + (1/Pi) * n/log(n+1) +- (...).

LINKS

Friedjof Tellkamp, Table of n, a(n) for n = 1..20000

Friedjof Tellkamp, Plots showing approximation and exact values for the first 3800 zeros

FORMULA

A374022(a(n)/2^(bigomega(a(n))-1)) ~ n.

EXAMPLE

The odd composite numbers (A071904) are: 9, 15, 21, 25, 27, ... .

Divide by 2^(bigomega()-1): 9/2, 15/2, 21/2, 25/2, 27/4, ... .

Sort: 9/2, 27/4, 15/2, 81/8, ... .

Take numerator: this sequence = 9, 27, 15, 81, ... .

MATHEMATICA

(*This algorithm calculates and sorts all noninteger rationals of the form x = k/2^(bigomega(k)-1) up to z, where the numerators of x are returned as the sequence*)

z = 100; op = Prime[Range[2, PrimePi[z]]]; (*A065091, odd primes*)

x = Select[Union[Flatten[Outer[Times, op, op/2]]], # <= z &];

For[i = 1, i < Max[1, Floor[Log[3/2, z/2]]] - 1, i++, x = Select[Union[x, Flatten[Outer[Times, x, op/2]]], # <= z &]]

a = Numerator[x] (*sequence*)

zzaprx = N@Table[2 Pi a[[i]]/2^PrimeOmega[a[[i]]] - i/2 + Sqrt[i]/2, {i, 1, Length[a]}] (*approximation for zeta zeros*)

CROSSREFS

Cf. A001222, A002410, A065091, A071904, A374022, A374603, A374604.