Feller–Tornier constant

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The Feller–Tornier constant, named after William Feller (1906 –1970) and Erhard Tornier (1894 –1982), is

CFeller–Tornier  =  1 2 1 +   ∞ ∏   i   = 1    1 − 2 pi  2  =  1 2 1 + 1 ζ (2)   ∞ ∏   i   = 1    1 − 1 pi  2 − 1  =

n → ∞limn → ∞  ∑ n i   = 1 [ω  (   i rad ( i  ) ) mod 2 = 0] n  ,

where

ω (n)

is the number of distinct prime factors of n,

rad (n)

is the radical of

n

,

ζ (2) = π 2 6

is zeta(2), and

[·]

is the Iverson bracket. The Feller–Tornier constant gives the density of natural numbers with an even number of distinct prime factors raised to powers larger than the first (ignoring any prime factors which appear only to the first power).

Note: A065474 defines the Feller–Tornier constant as (MISTAKE?) .

∞ ∏   i   = 1    1 − 2 pi  2   ,

where

pi

is the

i

th prime.

Decimal expansion

A065474 Decimal expansion of

∏ ∞ i   = 1 (1  −  2 / pi  2 )

= 0.322634098939244670579531692548...

{3, 2, 2, 6, 3, 4, 0, 9, 8, 9, 3, 9, 2, 4, 4, 6, 7, 0, 5, 7, 9, 5, 3, 1, 6, 9, 2, 5, 4, 8, 2, 3, 7, 0, 6, 6, 5, 7, 0, 9, 5, 0, 5, 7, 9, 6, 6, 5, 8, 3, 2, 7, 0, 9, 9, 6, 1, 8, 1, 1, ...}

A065493 Decimal expansion of

1 2 (1 + ∏ ∞ i   = 1 (1  −  2 / pi  2 ))

= 0.661317049469622335289765846274...

{6, 6, 1, 3, 1, 7, 0, 4, 9, 4, 6, 9, 6, 2, 2, 3, 3, 5, 2, 8, 9, 7, 6, 5, 8, 4, 6, 2, 7, 4, 1, 1, 8, 5, 3, 3, 2, 8, 5, 4, 7, 5, 2, 8, 9, 8, 3, 2, 9, 1, 6, 3, 5, 4, 9, 8, 0, 9, 0, 5, ...}

Continued fraction

The simple continued fraction for

∏ ∞ i   = 1 (1  −  2 / pi  2 )

gives the sequence of integer part and partial quotients (A074893, which omits the integer part 0)

{0, 3, 10, 19, 2, 1, 2, 2, 1, 6, 1, 6, 19, 17, 1, 7, 1, 2, 2, 1, 10, 2, 6, 2, 1, 3, 2, 1, 21, 5, 1, 15, 1, 1, 4, 1, 1, 1, 443, 2, 1, 4, 3, 1, 1, 6, 26, 6, 2, 39, 4, 1, 2, 6, 1, 1, 2, 4, ...}

The simple continued fraction for

1 2 (1 + ∏ ∞ i   = 1 (1  −  2 / pi  2 ))

gives the sequence of integer part and partial quotients (A078080)

{0, 1, 1, 1, 20, 9, 1, 2, 5, 1, 2, 3, 2, 3, 38, 8, 1, 16, 2, 2, 21, 1, 12, 1, 2, 1, 1, 2, 43, 2, 1, 32, 10, 3, 221, 1, 2, 9, 1, 1, 3, ...}

Density of integers with an even number of prime factors

The density of integers with an even number of prime factors is

n → ∞limn → ∞  ∑ n i  = 1 [Ω (i) mod 2 = 0] n  =  1 2 ,

where

Ω (n)

is the number of prime factors of n (with multiplicity) and

[⋅]

is the Iverson bracket. (This implies that the density of integers with an odd number of prime factors is also 1 / 2.) This is a consequence of the prime number theorem in the form

L (x) :=   n  ≤  x ∑   n  ≤  x    (−1) Ω (n)  =  o (x).

Density of integers with an even number of distinct prime factors

The density of integers with an even number of distinct prime factors is

n → ∞limn → ∞  ∑ n i  = 1 [ω (i) mod 2 = 0] n  =  ?,

where

ω (n)

is the number of distinct prime factors of n and

[·]

is the Iverson bracket.

Sequences

A007674 Numbers

n

such that

n

and

n + 1

are squarefree.

{1, 2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 46, 57, 58, 61, 65, 66, 69, 70, 73, 77, 78, 82, 85, 86, 93, 94, 101, 102, 105, 106, 109, 110, 113, 114, 118, 122, 129, ...}

A003557 n divided by the largest squarefree divisor of n.

{1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 2, 1, 1, 1, 8, 1, 3, 1, 2, 1, 1, 1, 4, 5, 1, 9, 2, 1, 1, 1, 16, 1, 1, 1, 6, 1, 1, 1, 4, 1, 1, 1, 2, 3, 1, 1, 8, 7, 5, 1, 2, 1, 9, 1, 4, 1, 1, 1, 2, ...}

External links

• Weisstein, Eric W., Feller-Tornier Constant, from MathWorld—A Wolfram Web Resource.