A336066 -id:A336066 - cp's OEIS frontend

A336067 Decimal expansion of the asymptotic density of numbers k divisible by A007814(k) (A336066).

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

Offset: 0

Amiram Eldar, Jul 07 2020

			0.435611028292172174213122545158086502293746907188068...

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. A007814, A336066, A336069.

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

Equals Sum_{k>=1} 1/(k * 2^(k - A007814(k) + 1)).

A336068 Numbers k such that the exponent of the highest power of 3 dividing k (A007949) is a divisor of k.

Original entry on oeis.org

3, 6, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 48, 51, 54, 57, 60, 66, 69, 72, 75, 78, 84, 87, 90, 93, 96, 102, 105, 108, 111, 114, 120, 123, 126, 129, 132, 135, 138, 141, 144, 147, 150, 156, 159, 165, 168, 174, 177, 180, 183, 186, 189, 192, 195, 198, 201, 204

Offset: 1

Amiram Eldar, Jul 07 2020

nonn

All the terms are divisible by 3 by definition.

Šalát (1994) proved that the asymptotic density of this sequence is 0.287106... (A336069).

			3 is a term since A007949(3) = 1 is a divisor of 3.

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.

A055777 is a subsequence.
Cf. A007949, A336069.
Similar sequences: A033950, A005349, A006446, A074946, A075592, A007694, A112249, A336066.

Select[Range[200], Mod[#, 3] == 0 && Divisible[#, IntegerExponent[#, 3]] &]

isok(m) = if (!(m%3), (m % valuation(m,3)) == 0); \\ Michel Marcus, Jul 08 2020

A370597 Even numbers k such that gcd(k, A007814(k)) = 1.

Original entry on oeis.org

2, 6, 8, 10, 14, 18, 22, 26, 30, 32, 34, 38, 40, 42, 46, 50, 54, 56, 58, 62, 66, 70, 74, 78, 82, 86, 88, 90, 94, 96, 98, 102, 104, 106, 110, 114, 118, 122, 126, 128, 130, 134, 136, 138, 142, 146, 150, 152, 154, 158, 162, 166, 170, 174, 178, 182, 184, 186, 190, 194, 198, 200

Offset: 1

Amiram Eldar, Feb 23 2024

The asymptotic density of this sequence is Sum_{k>=0} (phi(2*k+1)/((2*k+1)*2^(2*k+2))) = 0.30845704942203403516..., where phi is Euler's totient function (A000010).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A007814, A336066.

Select[2 * Range[200], CoprimeQ[#, IntegerExponent[#, 2]] &]

is(n) = !(n%2) && gcd(n, valuation(n, 2)) == 1;

cp's OEIS Frontend

A336067 Decimal expansion of the asymptotic density of numbers k divisible by A007814(k) (A336066).

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A336068 Numbers k such that the exponent of the highest power of 3 dividing k (A007949) is a divisor of k.

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A370597 Even numbers k such that gcd(k, A007814(k)) = 1.

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PARI