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.

2, 4, 12, 16, 32, 36, 112, 256, 72, 0, 180, 144, 216, 16384, 768, 65536, 432, 1600, 3072, 900, 864, 1296, 720, 12544, 1080, 67108864, 2592, 268435456, 1440, 9216, 196608, 5184, 2160, 17179869184, 2880, 36864, 10368, 3600, 6300

Offset: 0

Author

Juri-Stepan Gerasimov, Mar 05 2025

Keywords

• nonn

Comments

From Amiram Eldar, Mar 06 2025: (Start)

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.

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)

Crossrefs

Cf. A048105, A280892, A309181.

Extensions

a(25), a(27) and more terms from Amiram Eldar, Mar 06 2025