A164014 -id:A164014 - cp's OEIS frontend

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.

A369470 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, 2, 35, 455, 13624, 1176579

Offset: 1

Max Alekseyev, Jan 23 2024

For any n, a(n) >= A369469(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. A002966, A007850, A054377, A085098, A118086, A158649, A164014, A342267, A348625, A369469, A369607.

A075461 List of solutions to the Znám problem sorted first by length, then lexicographically.

Original entry on oeis.org

2, 3, 7, 47, 395, 2, 3, 11, 23, 31, 2, 3, 7, 43, 1823, 193667, 2, 3, 7, 47, 403, 19403, 2, 3, 7, 47, 415, 8111, 2, 3, 7, 47, 583, 1223, 2, 3, 7, 55, 179, 24323, 2, 3, 7, 43, 1807, 3263447, 2130014000915, 2, 3, 7, 43, 1807, 3263591, 71480133827, 2, 3, 7, 43

Offset: 1

Eric W. Weisstein, Sep 16 2002

			Starts with A075441(5)=2 5-term solutions 2,3,7,47,395; 2,3,11,23,31, followed by A075441(6)=5 6-term solutions, etc.

Max Alekseyev, Table of n, a(n) for n = 1..934 (flattened list of the n-term solutions for n = 5..8)
J. Janák and L. Skula, On the integers xi for which xi | x1 . . . xi-1 xi . . . xn + 1 holds, Mathematica Slovaca, Vol. 28 (1978), No. 3, 305--310.
J. Sondow and K. MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdos-Moser equation, Amer. Math. Monthly, 124 (2017) 232-240; arXiv:math/1812.06566 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Znám's Problem
Wikipedia, Znám's problem.

Cf. A006585, A007850, A054377, A075441, A085098, A118086, A158649, A164014, A343074, A369469.

Edited by Max Alekseyev, Jan 25 2024

A219186 Positive integers n such that 1+(k-2)*U_m(k,1)^2 does not divide n-k for any 3<=k=1, where U(k,1) is a Lucas sequence.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 14, 18, 20, 24, 32, 38, 42, 44, 54, 60, 62, 68, 72, 74, 80, 90, 98, 104, 108, 110, 114, 132, 140, 150, 152, 158, 164, 168, 180, 182, 194, 198, 200, 212, 234, 240, 242, 258, 270, 272, 278, 284, 294, 308, 312, 332, 338, 348, 350, 360, 368, 374, 380, 384, 398, 402, 410, 420, 422, 432, 434, 440, 450, 458, 464

Offset: 1

Max Alekseyev, Nov 14 2012

nonn

Related to solubility of generalized Markov equation x_1^2 + x_2^2 + ... + x_n^2 = k*x_1*x_2*...*x_n.

The sequence is infinite as proved by Baoulina and Luca.

I. Baoulina and F. Luca, On positive integers with a certain nondivisibility property, Annales Mathematicae et Informaticae 35 (2008), pp. 11-19.

Cf. A164014

cp's OEIS Frontend

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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A369470 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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A075461 List of solutions to the Znám problem sorted first by length, then lexicographically.

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A219186 Positive integers n such that 1+(k-2)*U_m(k,1)^2 does not divide n-k for any 3<=k=1, where U(k,1) is a Lucas sequence.

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