cp's OEIS Frontend

A325021 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820), tau(k) is the number of divisors of k (A000005), and sigma(k) is the sum of the divisors of k (A000203).

%I A325021 #25 May 10 2024 04:08:16
%S A325021 1,6,28,496,672,8128,30240,32760,332640,695520,2178540,17428320,
%T A325021 23569920,33550336,45532800,52141320,142990848,164989440,318729600,
%U A325021 447828480,481572000,500860800,540277920,623397600,644271264,714954240,995248800,1047254400,1307124000
%N A325021 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is an integer h, where k-tau(k) is the number of nondivisors of k (A049820), tau(k) is the number of divisors of k (A000005), and sigma(k) is the sum of the divisors of k (A000203).
%C A325021 Numbers m such that m*tau(m)/sigma(m) is an integer g and simultaneously m*(m-tau(m))/sigma(m) is an integer h. Corresponding values of integers g: 1, 2, 3, 5, 8, 7, 24, 24, 44, 46, 54, 96, 80, 13, 96, ...
%C A325021 Corresponding values of integers h: 0, 1, 11, 243, 216, 4057, 7536, 8166, 76186, 166589, ...
%C A325021 Even perfect numbers from A000396 are terms.
%C A325021 Complement of A325022 with respect to A001599.
%C A325021 Intersection of A325020 and A001599.
%H A325021 Amiram Eldar, <a href="/A325021/b325021.txt">Table of n, a(n) for n = 1..255</a> (terms below 10^14)
%e A325021 Harmonic number 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).
%t A325021 Select[Range[10^6], And[IntegerQ@ HarmonicMean@ #2, IntegerQ[#1 (#1 - #3)/#4]] & @@ Join[{#}, {Divisors@ #}, DivisorSigma[{0, 1}, #]] &] (* _Michael De Vlieger_, Mar 27 2019 *)
%o A325021 (Magma) [n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n) * n) / SumOfDivisors(n)) and IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n))]
%o A325021 (PARI) isok(m) = my(d=numdiv(m), s=sigma(m)); !frac(m*d/s) && !frac(m*(m-d)/s); \\ _Michel Marcus_, Mar 27 2019
%o A325021 (Python)
%o A325021 from itertools import count, islice
%o A325021 from math import prod
%o A325021 from functools import reduce
%o A325021 from sympy import factorint
%o A325021 def A325021_gen(startvalue=1): # generator of terms >= startvalue
%o A325021     for n in count(max(startvalue,1)):
%o A325021         f = factorint(n)
%o A325021         s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
%o A325021         if not (n*n%s or reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s):
%o A325021             yield n
%o A325021 A325021_list = list(islice(A325021_gen(),10)) # _Chai Wah Wu_, Feb 14 2023
%Y A325021 Cf. A000005, A000203, A001599, A007691, A049820, A325020, A325022, A325023, A325024.
%K A325021 nonn
%O A325021 1,2
%A A325021 _Jaroslav Krizek_, Mar 27 2019