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A325025 Numbers that are multi-perfect (A007691) and simultaneously harmonic (A001599). 4
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 (list; graph; refs; listen; history; text; internal format)
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
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
A325021 and A325023 are closely related sequences. - N. J. A. Sloane, May 03 2019
Sequence in context: A083387 A104511 A325021 * A325023 A138876 A326145
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, Mar 24 2019
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)