A374074
Odd composite numbers k sorted by k/2^(bigomega(k) - 1).
Original entry on oeis.org
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
Offset: 1
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, ... .
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(*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*)
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.
Original entry on oeis.org
9, 27, 15, 81, 21, 45, 25, 27, 243, 63, 33, 135, 35, 75, 39, 81, 45, 729, 189, 49, 99, 405, 51, 105, 55, 225, 57, 117, 243, 125, 63, 65, 135, 2187, 69, 567, 147, 297, 75, 1215, 153, 77, 315, 81, 165, 675, 85, 171, 87, 175, 351, 91, 729, 93, 375, 189, 95, 195
Offset: 1
k cannot be 1 or prime as this does not satisfy 1 < bigomega(k).
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, ... .
For i = 2, k is coprime to 2 and to 3, resulting in: 625/9, 875/9, 1225/9, ... , 15625/81, 15625/9, ... .
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, ... .
For i = 4 ... .
r(n) is the sorted union of the above subsequences.
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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 &];
Do[rat = Select[Union[Flatten[Outer[Times, rat, parr]]], # <= z &], id - 1];
While[ub >= irat^id, ub /= irat; parr = Select[parr, # <= ub &]; rat = Select[Union[rat, Flatten[Outer[Times, rat, parr/(id + 1)]]], # <= z &]];
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*)
rat = Select[rat, Mod[PrimeOmega[Numerator[#]], id] == 0 && Mod[PrimeOmega[Denominator[#]], id] == 0 &]; (*remove elements != 0 mod i*)
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, ...*)
rrtn = {}; imax = getimax[zmax]; For[i = 1, i <= imax, i++, rrtn = Union[rrtn, fi[i, zmax]]];
a = Numerator[rrtn]
Denominator[rrtn]; (*A374604*)
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