A140798 -id:A140798 - cp's OEIS frontend

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

Jaroslav Krizek, Mar 24 2019

nonn

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.

			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).

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

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

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

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

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

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

A325026 Multi-perfect numbers from A007691 that are not harmonic (A001599).

Original entry on oeis.org

120, 523776, 1476304896, 31998395520, 518666803200, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 69357059049509038080, 87934476737668055040, 170206605192656148480, 1161492388333469337600, 1802582780370364661760, 1940351499647188992000

Offset: 1

Jaroslav Krizek, Mar 24 2019

nonn

Multi-perfect numbers m such that m*tau(m)/sigma(m) is not an integer, where tau(k) is the number of the divisors of k (A000005) and sigma(k) is the sum of the divisors of k (A000203).
Supersequence of A046987.
Complement of A325025 with respect to A007691.

			120 is a term because 120*tau(120)/sigma(120) = 120*16/360 = 16/3.

Amiram Eldar, Table of n, a(n) for n = 1..202

Cf. A000005, A000203, A001599, A007691, A325025.

Cf. A140798 (harmonic numbers that are not multi-perfect).

[n: n in [1..1000000] | not IsIntegral((NumberOfDivisors(n)) * n / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]

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

cp's OEIS Frontend

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

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