A355626 -id:A355626 - cp's OEIS frontend

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).

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.
```

A355628 a(n) is the number of positive integers p that can be written as p = Product_{i = 1..n} (3 + 1/t_i) with integers t_i >= 2.

Original entry on oeis.org

0, 2, 7, 25, 96, 364, 1344, 4921

Offset: 1

Markus Sigg, Jul 15 2022

			a(1) = 0 because (3 + 1/t_1) is not integer for t_1 >= 2.
a(2) = 2 because 10 = (3 + 1/4) * (3 + 1/13) and 11 = (3 + 1/2) * (3 + 1/7) and no other positive integer p can be written as p = (3 + 1/t_1) * (3 + 1/t_2) with integers t_1, t_2 >= 2, cf. A355626.

Cf. A355626, A355627, 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.
```

A356275 a(n) is the number of tuples (t_1,t_2,m) of integers 2 <= t_1 <= t_2 and 0 < m < n such that (3 + 1/t_1)^m * (3 + 1/t_2)^(n-m) is an integer.

Original entry on oeis.org

3, 2, 4, 2, 5, 3, 5, 5, 5, 4

Offset: 2

Markus Sigg, Aug 03 2022

For each such tuple (t_1,t_2,m) we have t_1 < t_2 because (3 + 1/t)^n is not an integer for integer t >= 2.
Because for a positive integer t, no prime factor of t divides (3*t + 1), with p := (3 + 1/t_1)^m * (3 + 1/t_2)^(n-m) = (3*t_1 + 1)^m / t_1^m * (3*t_2 + 1)^(n-m) / t_2^(n-m) one sees that p is an integer iff t_1^m divides (3*t_2 + 1)^(n-m) and t_2^(n-m) divides (3*t_1 + 1)^m. Using this, the PARI program in the link calculates the first terms of the sequences A356275 - A356279. The program also uses that, because 3^n < p and so 3^n + 1 <= p <= (3 + 1/t_1)^n, there is the upper bound t_1 <= 1 / ((3^n + 1)^(1/n) - 3).
The pairs (t_1,t_2) that arise for integer products suggest this conjecture: For integers t_1,t_2 >= 2 and m,k > 0, the product (3 + 1/t_1)^m * (3 + 1/t_2)^k can be an integer only when (t_1,t_2) is one of (2,7), (2,7^2), (2^2,13), (2^2,13^3), (5,2^3).

			a(2) = 3: The tuples are (2,7,1), (4,13,1), (5,8,1) with (3 + 1/2)^1 * (3 + 1/7)^1 = 11 and (3 + 1/4)^1 * (3 + 1/13)^1 = (3 + 1/5)^1 * (3 + 1/8)^1 = 10.
a(3) = 2: The tuples are (2,49,2), (5,8,2) with (3 + 1/2)^2 * (3 + 1/49)^1 = 37 and (3 + 1/5)^2 * (3 + 1/8)^1 = 32.

Markus Sigg, PARI program.
Markus Sigg, Output of the PARI program.

Cf. A355626, A356276, A356277, A356278, A356279.

A356276 a(n) is the number of integers that can be written as (3 + 1/t_1)^m * (3 + 1/t_2)^(n-m) with integers t_1,t_2 >= 2 and 0 < m < n.

Original entry on oeis.org

2, 2, 3, 2, 4, 3, 4, 5, 4, 4

Offset: 2

Markus Sigg, Aug 03 2022

For comments and a PARI program see A356275.

Cf. A355626, A356275, A356277, A356278, A356279.

a(11) from Jinyuan Wang, Aug 04 2022

A356277 a(n) is the smallest integer that can be written as (3 + 1/t_1)^m * (3 + 1/t_2)^(n-m) with integers t_1,t_2 >= 2 and 0 < m < n.

Original entry on oeis.org

10, 32, 100, 320, 1000, 3125, 10000, 31250, 100000, 312500

Offset: 2

Markus Sigg, Aug 03 2022

For comments and a PARI program see A356275.

Cf. A355626, A356275, A356276, A356278, A356279.

a(11) from Jinyuan Wang, Aug 04 2022

A356278 a(n) is the largest integer that can be written as (3 + 1/t_1)^m * (3 + 1/t_2)^(n-m) with integers t_1,t_2 >= 2 and 0 < m < n.

Original entry on oeis.org

11, 37, 121, 325, 1369, 3250, 14641, 50653, 161051, 327680

Offset: 2

Markus Sigg, Aug 03 2022

For comments and a PARI program see A356275.

Cf. A355626, A356275, A356276, A356277, A356279.

a(11) from Jinyuan Wang, Aug 04 2022

A356279 Integers that can be written as (3 + 1/t_1)^m * (3 + 1/t_2)^k with integers t_1,t_2 >= 2 and m,k > 0.

Original entry on oeis.org

10, 11, 32, 37, 100, 103, 121, 320, 325, 1000, 1024, 1331, 1369, 3125, 3200, 3250, 10000, 10240, 10609, 14641, 31250, 32000, 32500, 32768, 50653, 100000, 102400, 105625, 161051, 312500, 320000, 325000, 327680

Offset: 1

Markus Sigg, Aug 03 2022

For comments and a PARI program see A356275.

Cf. A355626, A356275, A356276, A356277, A356278.

a(30)-a(33) from Jinyuan Wang, Aug 04 2022

cp's OEIS Frontend

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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A355628 a(n) is the number of positive integers p that can be written as p = Product_{i = 1..n} (3 + 1/t_i) with integers t_i >= 2.

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PARI

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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PARI

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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PARI

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

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PARI

A356275 a(n) is the number of tuples (t_1,t_2,m) of integers 2 <= t_1 <= t_2 and 0 < m < n such that (3 + 1/t_1)^m * (3 + 1/t_2)^(n-m) is an integer.

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A356276 a(n) is the number of integers that can be written as (3 + 1/t_1)^m * (3 + 1/t_2)^(n-m) with integers t_1,t_2 >= 2 and 0 < m < n.

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A356277 a(n) is the smallest integer that can be written as (3 + 1/t_1)^m * (3 + 1/t_2)^(n-m) with integers t_1,t_2 >= 2 and 0 < m < n.

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A356278 a(n) is the largest integer that can be written as (3 + 1/t_1)^m * (3 + 1/t_2)^(n-m) with integers t_1,t_2 >= 2 and 0 < m < n.

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A356279 Integers that can be written as (3 + 1/t_1)^m * (3 + 1/t_2)^k with integers t_1,t_2 >= 2 and m,k > 0.

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