A338395 -id:A338395 - cp's OEIS frontend

A338406 Numbers m such that tau(m), sigma(m) and pod(m) are pairwise relatively prime.

1, 4, 16, 25, 64, 81, 100, 121, 256, 289, 484, 529, 729, 841, 1024, 1156, 1296, 1600, 1681, 1936, 2116, 2209, 2401, 2809, 3025, 3364, 3481, 4096, 4624, 5041, 5184, 6400, 6724, 6889, 7225, 7921, 8464, 8836, 10201, 11236, 11449, 11664, 12100, 12769, 13225, 13456

Offset: 1

Jaroslav Krizek, Oct 24 2020

nonn

Numbers m such that A336723(m) = A000005(m) * A000203(m) * A007955(m).
Numbers m such that lcm(m, tau(m), sigma(m), pod(m)) = tau(m) * sigma(m) * pod(m).
Subsequence of numbers m such that A336722(m) = gcd(tau(m), sigma(m), pod(m)) = 1.
From David A. Corneth, Dec 11 2020: (Start)
a(n) is a perfect square. Proof: If a(n) is not a perfect square but is even then both tau(a(n)) and pod(a(n)) are divisible by 2. Contradiction.
If a(n) is not a perfect square and is odd then both tau(a(n)) and sigma(a(n)) are even. Contradiction.
Hence if a(n) is not a perfect square then it can be neither even nor odd. So a(n) is a perfect square. Q.E.D. (End)

			lcm(tau(4), sigma(4), pod(4)) = lcm(3, 7, 8) = tau(4) * sigma(4) * pod(4) = 3 * 7 * 8 = 168.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000005 (tau), A000203 (sigma), A007955 (pod).

Cf. A336722, A336723, A338395.

[m: m in [1..10^5] | LCM([#Divisors(m), &+Divisors(m), &*Divisors(m)]) eq #Divisors(m) * &+Divisors(m) * &*Divisors(m)]

Select[Range[15000], CoprimeQ[(d = DivisorSigma[0, #]), (s = DivisorSigma[1, #])] && CoprimeQ[d, (p = #^(d/2))] && CoprimeQ[s, p] &] (* Amiram Eldar, Oct 25 2020 *)

isok(m) = my(d=divisors(m), t=#d, s=vecsum(d), p=vecprod(d)); t*s*p == lcm([t,s,p]); \\ Michel Marcus, Oct 25 2020

cp's OEIS Frontend

A338406 Numbers m such that tau(m), sigma(m) and pod(m) are pairwise relatively prime.

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