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

1,2

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.

Links

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

Example

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).

Mathematica

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

Prog

(Magma) [n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n)) * n / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]
(PARI) isok(n) = my(s=sigma(n)); !frac(s/n) && !frac(n*numdiv(n)/s); \\ Michel Marcus, Mar 24 2019

Crossrefs

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

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

Sequence in context: A083387 A104511 A325021 * A325023 A138876 A326145

Adjacent sequences: A325022 A325023 A325024 * A325026 A325027 A325028

Keyword

nonn

Author

Jaroslav Krizek, Mar 24 2019

Status

approved