A333639 Numbers m such that g(m) = (m * tau(m) / sigma(m)), h(m) = (m * sigma(m)) / tau(m) and k(m) = (tau(m) * sigma(m)) / m are all integers.

1, 6, 672, 1638, 30240, 32760, 2178540, 17428320, 23569920, 29410290, 45532800, 714954240, 1379454720, 14182439040, 19209881600, 30600708096, 51001180160, 57575890944, 57629644800, 153003540480, 206166804480, 403031236608, 465036042240, 482476262400

Offset: 1

Author

Jaroslav Krizek, Mar 30 2020

Keywords

• nonn

Comments

Corresponding sequences of values of integers g(m), h(m) and k(m): {1, 2, 8, 9, 24, 24, 54, 96, 80, 81, 96, 200, ...}, {1, 18, 56448, 298116, 38102400, 44717400, 87889565400, 3164024354400, ...}, {1, 8, 72, 64, 384, 384, 864, 1944, 1280, 1024, 1536, 4608, 2304, 9600, 2916, ...}.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..34 (terms below 10^14)

Crossrefs

Subsequence of harmonic numbers (A001599).

Intersection of A001599, A333638 and A071707.

Cf. A000005, A000203.

Programs

Magma

[m: m in [1..10^5] | IsIntegral((m * #Divisors(m)) / &+Divisors(m)) and IsIntegral((&+Divisors(m) * m) / #Divisors(m)) and IsIntegral((&+Divisors(m) * #Divisors(m)) / m)]

Mathematica

Select[Range[10^5], Divisible[# * (d = DivisorSigma[0, #]), (s = DivisorSigma[1, #])] && Divisible[# * s, d] && Divisible[d * s, #] &] (* Amiram Eldar, Mar 31 2020 *)