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.

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

Author

Friedjof Tellkamp, Jul 13 2024

Keywords

• nonn
• frac

Comments

r(n) = a(n)/A374604(n).

r(n) is never an integer.

A374074(n)/2^(bigomega(A374074(n))-1) is a subsequence of r(n).

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.

Examples

			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.
		

Links

• Friedjof Tellkamp, Table of n, a(n) for n = 1..20000
• Friedjof Tellkamp, Plots showing approximation and exact values for the first 490000 zeta zeros

Crossrefs

Cf. A001222, A002410, A374074, A374604 (denominators).

Programs

Mathematica

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*)