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 A374074 Odd composite numbers k sorted by k/2^(bigomega(k) - 1). 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS Sorting by k/2^bigomega(k) would give the same sequence. 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. 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) +- (...). LINKS Friedjof Tellkamp, Table of n, a(n) for n = 1..20000 Friedjof Tellkamp, Plots showing approximation and exact values for the first 3800 zeros FORMULA A374022(a(n)/2^(bigomega(a(n))-1)) ~ n. EXAMPLE 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, ... . MATHEMATICA (*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*) CROSSREFS Cf. A001222, A002410, A065091, A071904, A374022, A374603, A374604. Sequence in context: A109041 A227900 A010817 * A205973 A122985 A255622 Adjacent sequences: A374071 A374072 A374073 * A374075 A374076 A374077 KEYWORD nonn AUTHOR Friedjof Tellkamp, Jun 27 2024 STATUS approved

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Last modified September 7 19:14 EDT 2024. Contains 375749 sequences. (Running on oeis4.)