A325025 Numbers that are multi-perfect (A007691) and simultaneously harmonic (A001599).

1, 6, 28, 496, 672, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 459818240, 1379454720, 8589869056, 14182439040, 43861478400, 51001180160, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 13661860101120

Offset: 1

Author

Jaroslav Krizek, Mar 24 2019

Keywords

• nonn

Comments

Multi-perfect numbers from A007691 that are harmonic numbers (A001599). Complement of A325026 with respect to A001599.

Harmonic numbers from A001599 that are multi-perfect numbers (A007691). Complement of A140798 with respect to A007691.

Numbers m such that sigma(m)/m is an integer g and simultaneously m*tau(m)/sigma(m) is an integer h, where tau(k) is the number of the divisors of k (A000005) and sigma(k) is the sum of the divisors of k (A000203). Corresponding values of integers g: 1, 2, 2, 2, 3, 2, 4, 4, 4, 4, 2, 4, 4, 3, 4, 2, 5, ... Corresponding values of integers h: 1, 2, 3, 5, 8, 7, 24, 24, 54, 80, 13, 96, 120, ...

Even perfect numbers from A000396 are terms.

Examples

			28 is a term because 28*tau(28)/sigma(28) = 28*6/56 = 3 (integer) and simultaneously 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..528

Crossrefs

Cf. A000005, A000203, A000396, A001599, A007691, A140798, A325026.

A325021 and A325023 are closely related sequences. - N. J. A. Sloane, May 03 2019

Programs

Magma

[n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n)) * n / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]

Mathematica

Select[Range[10^6], And[Mod[DivisorSigma[1, #], #] == 0, IntegerQ@ HarmonicMean@ Divisors@ #] &] (* Michael De Vlieger, Mar 24 2019 *)

PARI

isok(n) = my(s=sigma(n)); !frac(s/n) && !frac(n*numdiv(n)/s); \\ Michel Marcus, Mar 24 2019