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A374604 Denominators 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 A374604 #10 Jan 17 2025 19:41:57
%S A374604 2,4,2,8,2,4,2,2,16,4,2,8,2,4,2,4,2,32,8,2,4,16,2,4,2,8,2,4,8,4,2,2,4,
%T A374604 64,2,16,4,8,2,32,4,2,8,2,4,16,2,4,2,4,8,2,16,2,8,4,2,4,2,8,128,4,2,
%U A374604 32,8,2,16,4,64,8,2,4,2,16,2,2,4,4,2,8,2
%N A374604 Denominators 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 A374604 r(n) = A374603(n)/a(n).
%H A374604 Friedjof Tellkamp, <a href="/A374604/b374604.txt">Table of n, a(n) for n = 1..20000</a>
%e A374604 An example is given in A374603 (numerators corresponding to this sequence).
%t A374604 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 A374604 Do[rat = Select[Union[Flatten[Outer[Times, rat, parr]]], # <= z &], id - 1];
%t A374604 While[ub >= irat^id, ub /= irat; parr = Select[parr, # <= ub &]; rat = Select[Union[rat, Flatten[Outer[Times, rat, parr/(id + 1)]]], # <= z &]];
%t A374604 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 A374604 rat = Select[rat, Mod[PrimeOmega[Numerator[#]], id] == 0 && Mod[PrimeOmega[Denominator[#]], id] == 0 &]; (*remove elements != 0 mod i*)
%t A374604 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 A374604 rrtn = {}; imax = getimax[zmax]; For[i = 1, i <= imax, i++, rrtn = Union[rrtn, fi[i, zmax]]];
%t A374604 Numerator[rrtn] (*A374603*);
%t A374604 a = Denominator[rrtn]
%Y A374604 Cf. A001222, A002410, A374074, A374603 (numerators).
%K A374604 nonn,frac
%O A374604 1,1
%A A374604 _Friedjof Tellkamp_, Jul 13 2024