cp's OEIS Frontend

A381731 a(n) is the least number k with squarefree neighbors such that the number of non-unitary divisors of k (A048105) is equal to n, or 0 if no such k exists.

%I A381731 #28 Mar 25 2025 22:34:41
%S A381731 2,4,12,16,32,36,112,256,72,0,180,144,216,16384,768,65536,432,1600,
%T A381731 3072,900,864,1296,720,12544,1080,67108864,2592,268435456,1440,9216,
%U A381731 196608,5184,2160,17179869184,2880,36864,10368,3600,6300
%N A381731 a(n) is the least number k with squarefree neighbors such that the number of non-unitary divisors of k (A048105) is equal to n, or 0 if no such k exists.
%C A381731 From _Amiram Eldar_, Mar 06 2025: (Start)
%C A381731 For odd k a(k) is a square. a(9) = 0 because for a square m we have tau(m) >= 3^omega(m). Since A048105(m) = tau(m) - 2^omega(m) = 9, we have 2^omega(m) + 9 >= 3^omega(m) so omega(m) = 1.
%C A381731 Because m^2-1 is squarefree, m must be even, so with omega(m) = 1, we have m = 2^k and with tau(2^k) = 2^1 + 9 = 11 we get k = 10, m = 1024. But 1025 is not squarefree. Therefore a(9) = 0. (End)
%Y A381731 Cf. A048105, A280892, A309181.
%K A381731 nonn
%O A381731 0,1
%A A381731 _Juri-Stepan Gerasimov_, Mar 05 2025
%E A381731 a(25), a(27) and more terms from _Amiram Eldar_, Mar 06 2025