A325023 Multi-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

1, 6, 28, 496, 672, 8128, 30240, 32760, 2178540, 23569920, 33550336, 45532800, 142990848, 1379454720, 8589869056, 14182439040, 43861478400, 66433720320, 137438691328, 153003540480, 403031236608, 704575228896, 13661860101120, 181742883469056, 6088728021160320

Offset

1,2

Comments

Numbers m such that sigma(m)/m is an integer f and simultaneously m*tau(m)/sigma(m) is an integer g. Corresponding values of integers f: 1, 2, 2, 2, 3, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, ... Corresponding values of integers g: 0, 1, 11, 243, 216, 4057, 7536, 8166, ...

Complement of A325024 with respect to A007691.

Even perfect numbers from A000396 are terms.

Intersection of A325020 and A007691.

Conjecture: Numbers m such that all values of sigma(m)/m, m*tau(m)/sigma(m) and m*(m-tau(m))/sigma(m) are any integers (f, g, and h respectively). Corresponding values of integers f: 1, 2, 2, 2, 3, 2, 4, 4, 4, 4, 2, 4, 4, 4, 2, ... Corresponding values of integers g: 0, 1, 11, 243, 216, 4057, 7536, 8166, ... Corresponding values of integers h: 1, 2, 3, 5, 8, 7, 24, 24, 54, 80, 13, 96, ...

Links

Table of n, a(n) for n=1..25.

Example

Multi-perfect number 28 is a term because 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).

Mathematica

Select[Range[10^6], And[Mod[#3, #1] == 0, IntegerQ[#1 (#1 - #2)/#3]] & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* Michael De Vlieger, Mar 24 2019 *)

Prog

(Magma) [n: n in [1..1000000] | IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]
(PARI) isok(m) = my(s=sigma(m)); (frac(m*(m-numdiv(m))/s) == 0) && (frac(s/m) == 0); \\ Michel Marcus, Mar 25 2019

Crossrefs

Cf. A000005, A000203, A000396, A007691, A049820, A325020, A325021, A325022, A325024.

Sequence in context: A104511 A325021 A325025 * A138876 A326145 A174633

Adjacent sequences: A325020 A325021 A325022 * A325024 A325025 A325026

Keyword

nonn

Author

Jaroslav Krizek, Mar 24 2019

Status

approved