A051281 -id:A051281 - cp's OEIS frontend

A225362 Numbers n with the property that there are integers k, h such that sigma(n) = k^tau(n) = tau(n)^h.

1, 3, 217, 3937, 57337, 917497, 1040257, 11010027, 12189603, 16252897, 16646017, 3612185689, 4294434817, 66571993057, 68718821377, 17590038552577, 270545999761249, 281472829095937, 16138807601873739769, 16140901064495857657, 292842062170139131777, 1208766717309082486038529

Offset: 1

Jaroslav Krizek, May 05 2013

nonn

Intersection of A225239 and A051281.

			3937 is a term because sigma(3937) = 4096 = 8^tau(3937) = 8^4 = tau(3937)^6 = 4^6; k = 8, h = 6.

Cf. A000005 (tau=number of divisors), A000203 (sigma=sum of divisors).

Cf. A225239, A051281.

More terms, using A051281 b-file, from Michel Marcus, Feb 19 2020

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

Original entry on oeis.org

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

Jaroslav Krizek, Dec 16 2021

nonn

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)).

			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.

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

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

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

cp's OEIS Frontend

A225362 Numbers n with the property that there are integers k, h such that sigma(n) = k^tau(n) = tau(n)^h.

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