A350936 - OEIS

%I #20 Feb 20 2022 22:45:38

%S 34,6,12,30,816,60,192,270,180,240,56320,420,233472,2112,1620,1320,

%T 2162688,2340,786432,3120,4800,15360,62914560,3360,172368,724992,6300,

%U 29760,24964497408,12240,35433480192,7560,599040,15138816,81648,21600,7215545057280

%N a(n) is the smallest number m such that tau(m) = n*tau(m-1) = n*tau(m+1) or 0 if no such m exists, where tau(k) = A000005(k).

%C Corresponding values of tau(a(n)): 4, 4, 6, 8, 20, 12, 14, 16, 18, 20, 44, 24, 52, 28, 30, 32, 68, 36, 38, 40, 42, 44, 92, 48, 100, 52, 54, 56, 116, 60, 124, 64, 132, 136, 70, 72, 296, ...

%C Triples of [tau(a(n) - 1), tau(a(n)), tau(a(n) + 1)] = [tau(a(n)) / n, tau(a(n)), tau(a(n)) / n]: [4, 4, 4], [2, 4, 2], [2, 6, 2], [2, 8, 2], [4, 20, 4], [2, 12, 2], [2, 14, 2], [2, 16, 2], [2, 18, 2], [2, 20, 2], [4, 44, 4], ...

%e a(3) = 12 because 12 is the smallest number m such that tau(m) = 3 * tau(m-1) = 3 * tau(m+1); tau(12) = 3 * tau(11) = 3 * tau(13) = 3 * 2 = 6.

%o (Magma) Ax:=func<n|exists(r){m: m in[2..10^6] | n * #Divisors(m - 1) eq n * #Divisors(m + 1) and n * #Divisors(m + 1) eq #Divisors(m)} select r else 0>; [Ax(n): n in [1..16]]

%Y Cf. A000005, A190646, A350132, A350934, A350935.

%K nonn

%O 1,1

%A _Jaroslav Krizek_, Jan 25 2022

%E a(23)-a(37) from _Jon E. Schoenfield_, Jan 25 2022