A085098 -id:A085098 - cp's OEIS frontend

A118086 Number of ways 3/2 is a product of n superparticular ratios, without regard to order. A superparticular ratio is a ratio of the form m/(m-1).

1, 2, 14, 229, 9393, 1474291

Offset: 1

Gene Ward Smith, Apr 14 2006

The computations were made using Maple and a program of David Canright. The sequence relates to music theory, as 3/2 is the interval of a fifth in music and the sequence enumerates various ways of subdividing a fifth using superparticular intervals.

Cf. A085098.

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

A376012 a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = any integer.

Original entry on oeis.org

1, 1, 3, 12, 83, 1323, 63090, 14736464

Offset: 0

Keith F. Lynch, Sep 05 2024

Number of ways any integer is a product of n superparticular ratios, without regard to order. A superparticular ratio is a ratio of the form (m+1)/m.

			For n = 2, a(2) = 3, three solutions, {x_1, x_2} = {2, 3} = 2; {1, 2} = 3; {1, 1} = 4.
In other words, a(2) = 3 since 2 can be written as (3/2)(4/3), 3 can be written as (2/1)(3/2), and 4 can be written as (2/1)^2, but no other integers are the product of two superparticular ratios.
a(3) = 12 since 2 can be written in 5 ways, 3 can be written in 3 ways, and 4, 5, 6, and 8 can be written in 1 way each, as the product of three superparticular ratios.

Cf. A085098, A118086, A277608.

f(n, m, p, q)={ \\ the number of solutions in which product of the ratios is equal to p/q
    if(p<=q, return(0));
    if(n==1, return(if(q%(p-q)==0, 1, 0)));
    x=floor(1/((p/q)^(1/n)-1)); \\ x is the maximum of the least denominator of the ratios
    my(ans=0);
    for(i=m, x, ans=ans+f(n-1, i, p*i, q*(i+1)));
    \\ m indicates that the solutions require the denominators of all ratios not to be less than m
    \\ discard the ratio (i+1)/i
    return(ans);
};
a(n)={
    my(ans=0);
    for(i=2, 2^n, ans=ans+f(n, 1, i, 1));
    return(ans);
}; \\ Yifan Xie, Nov 21 2024

A376011 a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = 3.

Original entry on oeis.org

0, 1, 3, 19, 276, 10341, 1526969

Offset: 1

Keith F. Lynch, Sep 05 2024

Number of ways 3 is a product of n superparticular ratios, without regard to order. A superparticular ratio is a ratio of the form (m+1)/m.

			a(3) = 3 since 3 can be written as (2/1)(4/3)(9/8), (2/1)(5/4)(6/5), or (3/2)^2 (4/3) but in no other way using superparticular ratios.

Cf. A085098, A118086, A277608.

A334127 Number of nonempty sets {p_1, p_2, ..., p_k} such that Product_{i=1..k} p_i divides Product_{i=1..k} (n + p_i), where the p_i are distinct primes.

Original entry on oeis.org

1, 3, 4, 7, 6, 19, 8, 17, 8, 25, 12, 105, 8, 35, 22, 24, 16, 59, 28, 77, 30, 26, 22, 159, 8, 117, 23, 161, 26, 787, 32, 69, 46, 57, 30, 534, 32, 69, 90, 137, 22, 707, 20, 266, 54, 73, 50, 423, 38, 626, 62, 229, 52, 1324, 220, 489, 130, 285, 58, 2943, 24, 119, 274, 171, 202, 12089, 46, 181, 158, 201, 66, 1999, 58, 391, 234, 917, 126, 451, 42, 1767, 73, 1034, 86, 34691, 81, 150, 142, 233, 94, 18319, 226, 477, 70, 477, 114, 4419, 54, 157, 234, 2252

Offset: 1

Jinyuan Wang, May 10 2020

nonn

a(n) is always finite. Proof: let p_1 < p_2 < ... < p_k, we can prove p_k <= 2*n^2 + n. If p_k > 2*n^2 + n, then 2*p_k > p_k + n, thus p_k - n is in the set. If p_k - m*n is in the set and m < n, then 2*(p_k - m*n) > p_k + n, thus p_k - (m+1)*n is in the set. Therefore, p_k - m*n are in the set for 0 <= m <= n. Since p_k - n*n > n + 1, p_k - m*n can be divisible by n + 1 for some m <= n, which is a contradiction to the p_i being primes.

			For n = 3, {3}, {2, 3}, {2, 5} and {2, 3, 5} are such sets, thus a(3) = 4.

Cf. A085098, A118086, A355626, A355627, A355628, A355629, A355630, A355631, A376011, A376012,

a(n, k=primepi(2*n^2+n)) = {my(c=-1, p=primes(k)); forsubset(k, v, if(vecprod(vector(#v, i, p[v[i]]+n))%vecprod(vector(#v, i, p[v[i]])) == 0, c++)); c; }

Terms a(13) onward from Max Alekseyev, Apr 08 2025

cp's OEIS Frontend

A118086 Number of ways 3/2 is a product of n superparticular ratios, without regard to order. A superparticular ratio is a ratio of the form m/(m-1).

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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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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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A376012 a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = any integer.

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A376011 a(n) = number of solutions (x_1, x_2, ..., x_n) to Product_{i=1..n} (1 + 1/x_i) = 3.

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A334127 Number of nonempty sets {p_1, p_2, ..., p_k} such that Product_{i=1..k} p_i divides Product_{i=1..k} (n + p_i), where the p_i are distinct primes.

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