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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.

%I A356275 #21 Jan 12 2025 09:31:16
%S A356275 3,2,4,2,5,3,5,5,5,4
%N 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.
%C A356275 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.
%C A356275 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).
%C A356275 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).
%H A356275 Markus Sigg, <a href="/A356275/a356275.gp.txt">PARI program.</a>
%H A356275 Markus Sigg, <a href="/A356275/a356275.txt">Output of the PARI program.</a>
%e A356275 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.
%e A356275 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.
%Y A356275 Cf. A355626, A356276, A356277, A356278, A356279.
%K A356275 more,nonn
%O A356275 2,1
%A A356275 _Markus Sigg_, Aug 03 2022