A336068 -id:A336068 - cp's OEIS frontend

A336069 Decimal expansion of the asymptotic density of numbers k divisible by A007949(k) (A336068).

2, 8, 7, 1, 0, 6, 1, 3, 1, 8, 6, 6, 3, 4, 7, 1, 7, 3, 2, 2, 2, 8, 6, 1, 0, 3, 8, 3, 0, 0, 4, 9, 5, 5, 1, 0, 5, 9, 1, 4, 4, 3, 8, 1, 2, 3, 0, 9, 3, 8, 8, 9, 0, 6, 1, 0, 5, 5, 9, 2, 6, 9, 5, 7, 1, 0, 3, 8, 1, 4, 2, 1, 3, 2, 1, 0, 9, 1, 2, 0, 9, 0, 1, 9, 0, 3, 4

Offset: 0

Amiram Eldar, Jul 07 2020

			0.287106131866347173222861038300495510591443812309388...

Tibor Šalát, On the function a_p, p^a_p(n) || n (n > 1), Mathematica Slovaca, Vol. 44, No. 2 (1994), pp. 143-151.

Cf. A007949, A336067, A336068.

RealDigits[2 * Sum[1/k/3^(k + 1 - IntegerExponent[k, 3]), {k, 1, 1000}], 10, 100][[1]]

Equals 2 * Sum_{k>=1} 1/(k * 3^(k - A007949(k) + 1)).

A336066 Numbers k such that the exponent of the highest power of 2 dividing k (A007814) is a divisor of k.

Original entry on oeis.org

2, 4, 6, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 34, 36, 38, 42, 44, 46, 48, 50, 52, 54, 58, 60, 62, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 90, 92, 94, 98, 100, 102, 106, 108, 110, 112, 114, 116, 118, 120, 122, 124, 126, 130, 132, 134, 138, 140, 142, 144

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

All the terms are even by definition.
If m is a term then m*(2*k+1) is a term for all k>=1.
Šalát (1994) proved that the asymptotic density of this sequence is 0.435611... (A336067).

			2 is a term since A007814(2) = 1 is a divisor of 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Tibor Šalát, On the function a_p, p^a_p(n) || n (n > 1), Mathematica Slovaca, Vol. 44, No. 2 (1994), pp. 143-151.

A001146 and A039956 are subsequences.
Cf. A007814, A336067.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336068.

Select[Range[2, 150, 2], Divisible[#, IntegerExponent[#, 2]] &]

isok(m) = if (!(m%2), (m % valuation(m,2)) == 0); \\ Michel Marcus, Jul 08 2020

from itertools import count, islice
def A336066_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:n%(~n&n-1).bit_length()==0,count(max(startvalue+startvalue&1,2),2))
A336066_list = list(islice(A336066_gen(startvalue=3),30)) # Chai Wah Wu, Jul 10 2022

cp's OEIS Frontend

A336069 Decimal expansion of the asymptotic density of numbers k divisible by A007949(k) (A336068).

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