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A374603 Numerators of sorted rationals r(n) of the form k/d, where d=(i+1)^m, 1 <= m < bigomega(k), bigomega(k) == 0 (mod i), bigomega(d) == 0 (mod i) and gcd(k, prime(j)) = 1 for all j <= i.

%I A374603 #12 Jan 17 2025 19:40:35
%S A374603 9,27,15,81,21,45,25,27,243,63,33,135,35,75,39,81,45,729,189,49,99,
%T A374603 405,51,105,55,225,57,117,243,125,63,65,135,2187,69,567,147,297,75,
%U A374603 1215,153,77,315,81,165,675,85,171,87,175,351,91,729,93,375,189,95,195
%N A374603 Numerators of sorted rationals r(n) of the form k/d, where d=(i+1)^m, 1 <= m < bigomega(k), bigomega(k) == 0 (mod i), bigomega(d) == 0 (mod i) and gcd(k, prime(j)) = 1 for all j <= i.
%C A374603 r(n) = a(n)/A374604(n).
%C A374603 r(n) is never an integer.
%C A374603 A374074(n)/2^(bigomega(A374074(n))-1) is a subsequence of r(n).
%C A374603 Conjecture: It appears that Pi*r(n) + sqrt(n) ~~ A002410(n), where ‘~~’ means ‘close to’. The relative error is less than +-0.001 for n ranging from 100000 to 400000.
%H A374603 Friedjof Tellkamp, <a href="/A374603/b374603.txt">Table of n, a(n) for n = 1..20000</a>
%H A374603 Friedjof Tellkamp, <a href="/A374603/a374603.png">Plots showing approximation and exact values for the first 490000 zeta zeros</a>
%e A374603 k cannot be 1 or prime as this does not satisfy 1 < bigomega(k).
%e A374603 For i = 1, k is an odd composite number, resulting in (unsorted) k/d: 9/2, 15/2, 21/2, 25/2, 27/4, 27/2, ... , 81/8, 81/4, 81/2, ... .
%e A374603 For i = 2, k is coprime to 2 and to 3, resulting in: 625/9, 875/9, 1225/9, ... , 15625/81, 15625/9, ... .
%e A374603 For i = 3, k is coprime to 2, to 3 and to 5, resulting in: 7^6/4^3, (11*7^5)/4^3, ... , 7^9/4^6, ... .
%e A374603 For i = 4 ... .
%e A374603 r(n) is the sorted union of the above subsequences.
%t A374603 zmax = 200; fi[id_, z_] := (irat = (id + 2)/(id + 1); ub = z/irat^id; parr = Select[Prime[Range[id + 1, PrimePi[z]]], # <= ub &]; rat = Select[Union[Flatten[Outer[Times, parr, parr]]]/(id + 1), # <= z &];
%t A374603 Do[rat = Select[Union[Flatten[Outer[Times, rat, parr]]], # <= z &], id - 1];
%t A374603 While[ub >= irat^id, ub /= irat; parr = Select[parr, # <= ub &]; rat = Select[Union[rat, Flatten[Outer[Times, rat, parr/(id + 1)]]], # <= z &]];
%t A374603 iw = 1; While[iw <= Length[rat], If[Denominator[rat[[iw]]] >= (id + 1)^2 && (id + 1) rat[[iw]] <= z, AppendTo[rat, (id + 1) rat[[iw]]]]; iw++]; (*append multiples of k/d*)
%t A374603 rat = Select[rat, Mod[PrimeOmega[Numerator[#]], id] == 0 && Mod[PrimeOmega[Denominator[#]], id] == 0 &]; (*remove elements != 0 mod i*)
%t A374603 Return[Union[rat]]; ); getimax[zi_] := (im = 1; While[Prime[im + 1]^(2 im)/(im + 1)^im <= zi, im++]; Return[Max[1, im - 1]]); (*1 for z<625/9, 2 for z<7^6/4^3, ...*)
%t A374603 rrtn = {}; imax = getimax[zmax]; For[i = 1, i <= imax, i++, rrtn = Union[rrtn, fi[i, zmax]]];
%t A374603 a = Numerator[rrtn]
%t A374603 Denominator[rrtn]; (*A374604*)
%Y A374603 Cf. A001222, A002410, A374074, A374604 (denominators).
%K A374603 nonn,frac
%O A374603 1,1
%A A374603 _Friedjof Tellkamp_, Jul 13 2024