A374022 -id:A374022 - cp's OEIS frontend

A374074 Odd composite numbers k sorted by k/2^(bigomega(k) - 1).

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

Friedjof Tellkamp, Jun 27 2024

nonn

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) +- (...).

			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, ... .

Friedjof Tellkamp, Table of n, a(n) for n = 1..20000
Friedjof Tellkamp, Plots showing approximation and exact values for the first 3800 zeros

Cf. A001222, A002410, A065091, A071904, A374022, A374603, A374604.

A373943 a(n) is the cardinality of the set containing all rational numbers of the form 2 <= m/2^(bigomega(m) - 1) <= n.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 6, 7, 7, 7, 10, 11, 13, 13, 13, 15, 18, 19, 21, 22, 22, 22, 24, 25, 27, 29, 30, 31, 34, 35, 36, 37, 38, 38, 40, 41, 43, 45, 47, 48, 49, 50, 54, 57, 57, 58, 61, 62, 63, 63, 63, 65, 66, 67, 67, 70, 71, 74, 75, 77, 79, 82, 82, 84, 86, 89, 91

Offset: 1

Friedjof Tellkamp, Jun 23 2024

nonn

			a(10) = 7 = card{2, 3, 9/2, 5, 27/4, 7, 15/2}.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000720, A001222, A052130, A374022.

z = 100;
k[n_] := Max[1, Floor[Log[3/2, n/2]]];
m[n_] := n 2^(k[n] - 1);
PrimePiK = Table[0, Floor[Log[2, m[z]]], m[z]];
For[i = 2, i <= m[z], i++, PrimePiK[[PrimeOmega[i], i]] = 1]
PrimePiK = Accumulate /@ PrimePiK;
a = Table[PrimePiK[[k[n], m[n]]], {n, z}] (*sequence*)
x = Union@Select[Table[i/2^(PrimeOmega[i] - 1), {i, 1, m[z], 2}], # <= z &] (*set*)

nap(n, k) = sum(i=1, n, bigomega(i)==k);
a(n) = my(k=max(1, floor(log(n/2)/(log(3)-log(2))))); nap(n*2^(k-1), k); \\ Michel Marcus, Jun 27 2024

from math import isqrt, prod
from sympy import primepi, primerange, integer_nthroot
def A373943(n):
    if n<=4: return primepi(n)
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    k = 1
    while 3**k<(r:=n<Chai Wah Wu, Dec 03 2024

a(n) = card{x | x = m/2^(bigomega(m)-1), x<=n}.
a(n) = pi_k(n * 2^(k - 1)), with pi_k(n) as the counting function for k-almost primes and k sufficiently large.
k needs to be at least max(1, floor(log(n/2)/(log(3)-log(2)))) and m = n * 2^(k - 1).
a(n) = A374022(n) + A000720(n).
a(2^n) = A052130(n-1).

cp's OEIS Frontend

A374074 Odd composite numbers k sorted by k/2^(bigomega(k) - 1).

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