A343945 a(1) = 1; for n >= 2, a(n) = floor(log(sigma(n)) / log(tau(n))).

1, 1, 2, 1, 2, 1, 3, 1, 2, 2, 3, 1, 3, 2, 2, 2, 4, 2, 4, 2, 2, 2, 4, 1, 3, 2, 2, 2, 4, 2, 5, 2, 2, 2, 2, 2, 5, 2, 2, 2, 5, 2, 5, 2, 2, 3, 5, 2, 3, 2, 3, 2, 5, 2, 3, 2, 3, 3, 5, 2, 5, 3, 2, 2, 3, 2, 6, 2, 3, 2, 6, 2, 6, 3, 2, 2, 3, 2, 6, 2, 2, 3, 6, 2, 3, 3, 3

Offset: 1

Author

Jaroslav Krizek, Dec 16 2021

Keywords

• nonn

Comments

See A051281 for numbers m such that sigma(m) = tau(m)^k where k = integer, i.e., numbers m such that floor(log(sigma(m)) / log(tau(m))) = log(sigma(m)) / log(tau(m)).

Examples

			a(6) = floor(log(sigma(6)) / log(tau(6))) = floor(log(12) / log(4)) = floor(1.792481...) = 1.
a(7) = floor(log(sigma(7)) / log(tau(7))) = floor(log(8) / log(2)) = floor(3) = 3.
		

Crossrefs

Cf. A000005(tau), A000203(sigma), A051281, A334455, A349006, A349007.

Programs

Magma

[1] cat [Floor(Log(&+Divisors(n)) / Log(#Divisors(n))): n in [2..100]]

Mathematica

a[1] = 1; a[n_] := Floor[Log @@ DivisorSigma[{0, 1}, n]]; Array[a, 100] (* Amiram Eldar, Dec 16 2021 *)