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A374074 Odd composite numbers k sorted by k/2^(bigomega(k) - 1).

%I A374074 #34 Dec 13 2024 14:17:50
%S A374074 9,27,15,81,21,45,25,243,63,33,135,35,75,39,729,189,49,99,405,51,105,
%T A374074 55,225,57,117,125,65,2187,69,567,147,297,1215,153,77,315,165,675,85,
%U A374074 171,87,175,351,91,93,375,95,195,6561,207,1701,441,111,891,3645,459
%N A374074 Odd composite numbers k sorted by k/2^(bigomega(k) - 1).
%C A374074 Sorting by k/2^bigomega(k) would give the same sequence.
%C A374074 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.
%C A374074 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) +- (...).
%H A374074 Friedjof Tellkamp, <a href="/A374074/b374074.txt">Table of n, a(n) for n = 1..20000</a>
%H A374074 Friedjof Tellkamp, <a href="/A374074/a374074_1.jpg">Plots showing approximation and exact values for the first 3800 zeros</a>
%F A374074 A374022(a(n)/2^(bigomega(a(n))-1)) ~ n.
%e A374074 The odd composite numbers (A071904) are: 9, 15, 21, 25, 27, ... .
%e A374074 Divide by 2^(bigomega()-1): 9/2, 15/2, 21/2, 25/2, 27/4, ... .
%e A374074 Sort: 9/2, 27/4, 15/2, 81/8, ... .
%e A374074 Take numerator: this sequence = 9, 27, 15, 81, ... .
%t A374074 (*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*)
%t A374074 z = 100; op = Prime[Range[2, PrimePi[z]]]; (*A065091, odd primes*)
%t A374074 x = Select[Union[Flatten[Outer[Times, op, op/2]]], # <= z &];
%t A374074 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 &]]
%t A374074 a = Numerator[x] (*sequence*)
%t A374074 zzaprx = N@Table[2 Pi a[[i]]/2^PrimeOmega[a[[i]]] - i/2 + Sqrt[i]/2, {i, 1, Length[a]}] (*approximation for zeta zeros*)
%Y A374074 Cf. A001222, A002410, A065091, A071904, A374022, A374603, A374604.
%Y A374074 See also A122943.
%K A374074 nonn
%O A374074 1,1
%A A374074 _Friedjof Tellkamp_, Jun 27 2024