A369470 -id:A369470 - cp's OEIS frontend

A342267 Number of partitions of 1/n into n reciprocals of positive integers.

1, 2, 21, 694, 118995, 132891609

Offset: 1

Ilya Gutkovskiy, Mar 07 2021

			a(2) = 2 because we have 1/2 = 1/4 + 1/4 = 1/3 + 1/6.

Cf. A002966, A003167, A004194, A018892, A020327, A020328, A343074, A343214, A369470.

a(6) from Jud McCranie, Sep 02 2021

A369469 a(n) = number of integer solutions to 1 <= x1 < x2 < ... < xn to 1/x1 + ... + 1/xn = (1 - 1/x1) * ... * (1 - 1/xn).

Original entry on oeis.org

1, 1, 1, 24, 293, 9219, 787444

Offset: 1

Max Alekseyev, Jan 23 2024

For any n, A369470(n) >= a(n) >= 1 (see A369607).

Max Muller et al., The diophantine equation \sum_{n=1}^{N} \frac{1}{x_{n}} = \prod_{k=1}^{N} \left(1-\frac{1}{x_{k}} \right), MathOverflow, 2024.

Cf. A006585, A007850, A054377, A085098, A118086, A158649, A164014, A343074, A369470, A369607.

A369607 Greedy solution a(1) < a(2) < ... to 1/a(1) + 1/a(2) + ... = (1 - 1/a(1)) * (1 - 1/a(2)) * ....

Original entry on oeis.org

3, 6, 29, 803, 643727, 414383582243, 171713753231982206218247, 29485613049014079571725771288849499850026859243, 869401376876189366008603664962520703088459987798626788985159595026678611496977754082506135887

Offset: 1

Max Alekseyev, Jan 27 2024

nonn

For any n, (x1, x2, ..., xn) = (a(1), a(2), ..., a(n-1), a(n)-1) forms a solution to 1/x1 + ... + 1/xn = (1 - 1/x1) * ... * (1 - 1/xn), proving that A369470(n) >= A369469(n) >= 1.

Max Muller et al., The diophantine equation Sum_{n=1..N} 1/x_n = Product_{k=1..N} (1-1/x_k), MathOverflow, 2024.

Cf. A000058, A348625, A348626, A369469, A369470.

a(n+2) = a(n+1)^2 + (a(n) - 2)*a(n+1) - a(n)^3 + 2*a(n)^2 - 2*a(n) + 2.

cp's OEIS Frontend

A342267 Number of partitions of 1/n into n reciprocals of positive integers.

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A369469 a(n) = number of integer solutions to 1 <= x1 < x2 < ... < xn to 1/x1 + ... + 1/xn = (1 - 1/x1) * ... * (1 - 1/xn).

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A369607 Greedy solution a(1) < a(2) < ... to 1/a(1) + 1/a(2) + ... = (1 - 1/a(1)) * (1 - 1/a(2)) * ....

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