A356277 -id:A356277 - cp's OEIS frontend

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.

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

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

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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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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