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a(n) is the number of completely reduced tower fractions in a 2^n X 2^n X 2^n X 2^n division tesseract matrix.

Number of ordered quadruples (w, x, y, z) with gcd(w, x, y, z) = 1 and 1 <= {w, x, y, z} <= 2^n.

Only tower fractions with gcd(x, y, z, w) = 1 are counted.

The following sequences are closely related:

A018805 (1/zeta(2) in n steps);

A342632 (1/zeta(2) in 2^n steps);

A342586 (1/zeta(2) in 10^n steps);

A071778 (1/zeta(3) in n steps);

A342935 (1/zeta(3) in 2^n steps);

A342841 (1/zeta(3) in 10^n steps);

A082540 (1/zeta(4) in n steps);

here (1/zeta(4) in 2^n steps);

A343193 (1/zeta(4) in 10^n steps).

Joachim von zur Gathen and Jürgen Gerhard, Modern Computer Algebra, Cambridge University Press, Second Edition 2003, pp. 53-54.

Robin Whitty, <a href="https://www.theoremoftheday.org/LogicAndComputerScience/CountableQ/TotDCantorQ.pdf">Countability of the Rationals</a>, Theorem of the day (2021).

Wikipedia, <a href="https://en.wikipedia.org/wiki/Farey_sequence">Farey Sequence</a>.

Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_function">Arithmetic function</a>.

Wikipedia, <a href="https://en.wikipedia.org/wiki/Tesseract">Tesseract</a>.

Lim_{n->infinity} 2^(4*n)/a(n) = zeta(4) = A013662 = Pi^4/90.

1/2/2/3 counts, but 2/4/4/6, 3/6/6/9 ... do not count, because they reduce to 1/2/2/3;

1/1/1/1 counts, but 2/2/2/2, 3/3/3/3 ... do not count, because they reduce to 1/1/1/1.

For n=3, the size of the division tesseract matrix gris is 8 X 8 X 8 X 8:

Edited by N. J. A. Sloane, Jun 13 2021

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Chai Wah Wu, <a href="/A343527/b343527_1.txt">Table of n, a(n) for n = 0..52</a> (n = 0..31 from Karl-Heinz Hofmann)

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Karl-Heinz Hofmann, Chai Wah Wu, <a href="/A343527/b343527_1.txt">Table of n, a(n) for n = 0..52</a> (n = 0..31</a> from Karl-Heinz Hofmann)

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