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

1,2

Comments

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)

Links

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

Example

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

Mathematica

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

Prog

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

Crossrefs

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

Cf. A336722, A336723, A338395.

Sequence in context: A353295 A363428 A295921 * A351979 A111350 A223221

Adjacent sequences: A338403 A338404 A338405 * A338407 A338408 A338409

Keyword

nonn

Author

Jaroslav Krizek, Oct 24 2020

Status

approved