A355628 -id:A355628 - cp's OEIS frontend

A355626 a(n) is the number of tuples (t_1, ..., t_n) with integers 2 <= t_1 <= ... <= t_n such that Product_{i = 1..n} (3 + 1/t_i) is an integer.

0, 3, 80, 15222

Offset: 1

Markus Sigg, Jul 15 2022

Bounds on the values t_i can be derived as shown in "Bounds on t_i when the product of factors (3 + 1/t_i) is given" in the Links section. These bounds allow to calculate a(3) = 80 and a(4) = 15222 with the PARI program in the link. It seems that determining a(5) would need stronger methods. The identity (3 + 1/82) * (3 + 1/6670) * (3 + 1/44484454) * (3 + 1/1978866618021814) * (3 + 1/3915913091921090597566167836053) = 244 shows that the values t_n can get quite large. Integer products of more factors (3 + 1/t_i) can have even much larger t_n.

			a(1) = 0: As 1/t_1 is not an integer for t_1 >= 2, there is no t_1 >= 2 with integer 3 + 1/t_1.
a(2) = 3: With p := (3 + 1/t_1) * (3 + 1/t_2) we have p > 9, so for integer p it is p >= 10. With p <= (3 + 1/t_1)^2 we get t_1 <= 6. Solving p = 10, p = 11, p = 12 with 2 <= t_1 <= 6 for t_2 shows that the only integer solutions are (t_1,t_2) = (4,13) and (t_1,t_2) = (5,8) for p = 10, and (t_1,t_2) = (2,7) for p = 11.

Markus Sigg, Bounds on t_i when the product of factors (3 + 1/t_i) is given
Markus Sigg, PARI program.

Cf. A355627, A355628, A355629, A355630, A355631.

\\ See link. The program has code for sequences A355626, ..., A355631 and can handle the more general case of factors (s + 1/t_i) with s >= 2.

A355627 a(n) is the number of tuples (t_1, ..., t_k) with a positive integer k and integers 2 <= t_1 <= ... <= t_k such that n = Product_{i = 1..k} (3 + 1/t_i).

Original entry on oeis.org

2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 50, 14, 0, 2, 9, 0, 2, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 9291, 1668, 0, 2170, 226, 0, 1052, 59, 0

Offset: 10

Markus Sigg, Jul 15 2022

nonn

Because 3^k < Product_{i = 1..k} (3 + 1/t_i) < 3.5^k, a(n) > 0 is possible only for 10 <= n <= 12 (k = 2), 28 <= n <= 42 (k = 3), 82 <= n <= 150 (k = 4), 244 <= n <= 525 (k = 5) etc. For n <= 19683, there can exist at most one k such that n can be written as a product of k factors (3 + 1/t_i).
a(n) = 0 when n is a multiple of 3: Suppose n = Product_{i = 1..k} (3 + 1/t_i). Then n * Product_{i = 1..k} t_i = Product_{i = 1..k} (3 * t_i + 1). The right hand side is not a multiple of 3, so neither n nor any of the t_i can be a multiple of 3.
a(n) > 0 iff n is in A355631.

Markus Sigg, Table of n, a(n) for n = 10..243

Cf. A355626, A355628, A355629, A355630, A355631.

PARI
```
\\ See A355626.
```

A355629 a(n) is the number of tuples (t_1, ..., t_n) with integers 2 <= t_1 <= ... <= t_n such that 3^n + 1 = Product_{i = 1..n} (3 + 1/t_i).

Original entry on oeis.org

0, 2, 50, 9291

Offset: 1

Markus Sigg, Jul 15 2022

			a(2) = 2: 3^2 + 1 = 10 can be expressed as (3 + 1/4) * (3 + 1/13) and as (3 + 1/5) * (3 + 1/8);
a(3) = 50:  There are 50 representations of 3^3 + 1 = 28 with 10 <= min(t_i) <= 23 and 38 <= max(t_i) <= 8773. A product with minimal t_1 and maximal t_3 is 28 = (3 + 1/10) * (3 + 1/94) * (3 + 1/8773), maximal t_1 and minimal t_3 occur in 28 = (3 + 1/23) * (3 + 1/25) * (3 + 1/38).

Cf. A034472, A355626, A355627, A355628, A355630, A355631.

A356210 is the same problem with target 2^n + 1 and factors (2 + 1/t_k).

PARI
```
See A355626.
```

A355630 a(n) is the largest integer that can be written as Product_{i = 1..n} (3 + 1/t_i) with integers t_i >= 2.

Original entry on oeis.org

11, 37, 121, 413, 1442, 5047, 16807, 58457, 204085, 709667, 2483663, 8068753, 30415033

Offset: 2

Markus Sigg, Jul 15 2022

Obviously 3^n < a(n) < 3.5^n.

			11 = (3 + 1/2) * (3 + 1/7) is the largest integer p that can be written as p = (3 + 1/t_1) * (3 + 1/t_2) with integers t_1,t_2 >= 2 because any such integer p is smaller than 3.5^2 = 12.25 and there is no such representation for p = 12. Hence, a(2) = 11.

Cf. A355626, A355627, A355628, A355629, A355631.

PARI
```
See A355626.
```

A355631 List of numbers k such that A355627(k) > 0.

Original entry on oeis.org

10, 11, 28, 29, 31, 32, 34, 35, 37, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 103, 104, 106, 110, 112, 113, 115, 116, 118, 119, 121, 244, 245, 247, 248, 250, 251, 253, 254, 256, 257, 259, 260, 262, 263, 265, 266, 268, 269, 271, 272, 274, 275, 277, 278, 280

Offset: 1

Markus Sigg, Jul 15 2022

nonn

A positive integer p is in this list iff it can be written as p = Product_{i = 1..k} (3 + 1/t_i) with a positive integer k and integers t_i >= 2.

The sequence does not contain multiples of 3, see comments in A355627.

			10 = (3 + 1/4) * (3 + 1/13). 100 = (3 + 1/2) * (3 + 1/7) * (3 + 1/34) * (3 + 1/1133). 280 = (3 + 1/4) * (3 + 1/7) * (3 + 1/22) * (3 + 1/2614) * (3 + 1/6831253).

Markus Sigg, Table of n, a(n) for n = 1..13637

Cf. A001651, A355626, A355627, A355628, A355629, A355630.

PARI
```
See A355626.
```

A355516 a(n) is the number of distinct integer values of Product_{k=1..n} (2 + 1/t_k) with integers t_k > 1.

Original entry on oeis.org

1, 2, 5, 11, 29, 70, 164, 392, 933

Offset: 2

Hugo Pfoertner and Markus Sigg, Jul 16 2022

			a(2) = 1: The only representable integer is 5 = (2 + 1/3) * (2 + 1/7).
a(3) = 2: 9 and 11 can be represented; 9 = (2 + 1/9) * (2 + 1/13) * (2 + 1/19) and 10 other ways; 11 = (2 + 1/3)*(2 + 1/5)*(2 + 1/7) = (2 + 1/3)^2 * (2 + 1/49).

A355626 provides more information.
A355628 same problem with factors (3 + 1/t_k).
Cf. A355243.

PARI
```
\\ See link in A355626.
```

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

A355626 a(n) is the number of tuples (t_1, ..., t_n) with integers 2 <= t_1 <= ... <= t_n such that Product_{i = 1..n} (3 + 1/t_i) is an integer.

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A355627 a(n) is the number of tuples (t_1, ..., t_k) with a positive integer k and integers 2 <= t_1 <= ... <= t_k such that n = Product_{i = 1..k} (3 + 1/t_i).

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A355629 a(n) is the number of tuples (t_1, ..., t_n) with integers 2 <= t_1 <= ... <= t_n such that 3^n + 1 = Product_{i = 1..n} (3 + 1/t_i).

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A355630 a(n) is the largest integer that can be written as Product_{i = 1..n} (3 + 1/t_i) with integers t_i >= 2.

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A355631 List of numbers k such that A355627(k) > 0.

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A355516 a(n) is the number of distinct integer values of Product_{k=1..n} (2 + 1/t_k) with integers t_k > 1.

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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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