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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).

%I A325023 #17 May 12 2024 00:46:35
%S A325023 1,6,28,496,672,8128,30240,32760,2178540,23569920,33550336,45532800,
%T A325023 142990848,1379454720,8589869056,14182439040,43861478400,66433720320,
%U A325023 137438691328,153003540480,403031236608,704575228896,13661860101120,181742883469056,6088728021160320
%N 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).
%C A325023 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, ...
%C A325023 Complement of A325024 with respect to A007691.
%C A325023 Even perfect numbers from A000396 are terms.
%C A325023 Intersection of A325020 and A007691.
%C A325023 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, ...
%H A325023 Amiram Eldar, <a href="/A325023/b325023.txt">Table of n, a(n) for n = 1..512</a>
%e A325023 Multi-perfect number 28 is a term because 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).
%t A325023 Select[Range[10^6], And[Mod[#3, #1] == 0, IntegerQ[#1 (#1 - #2)/#3]] & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* _Michael De Vlieger_, Mar 24 2019 *)
%o A325023 (Magma) [n: n in [1..1000000] | IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n)) and IsIntegral(SumOfDivisors(n)/n)]
%o A325023 (PARI) isok(m) = my(s=sigma(m)); (frac(m*(m-numdiv(m))/s) == 0) && (frac(s/m) == 0); \\ _Michel Marcus_, Mar 25 2019
%Y A325023 Cf. A000005, A000203, A000396, A007691, A049820, A325020, A325021, A325022, A325024.
%K A325023 nonn
%O A325023 1,2
%A A325023 _Jaroslav Krizek_, Mar 24 2019