cp's OEIS Frontend

A325022 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is not an integer, where k-tau(k) = the number of nondivisors of k (A049820), tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

%I A325022 #24 May 10 2024 04:08:12
%S A325022 140,270,1638,2970,6200,8190,18600,18620,27846,55860,105664,117800,
%T A325022 167400,173600,237510,242060,360360,539400,726180,753480,950976,
%U A325022 1089270,1421280,1539720,2229500,2290260,2457000,2845800,4358600,4713984,4754880,5772200,6051500
%N A325022 Harmonic numbers m from A001599 such that m*(m-tau(m))/sigma(m) is not an integer, where k-tau(k) = the number of nondivisors of k (A049820), tau(k) = the number of divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).
%C A325022 Numbers m such that sigma(m) divides m*tau(m) but sigma(m) does not divide m*(m-tau(m)).
%C A325022 Complement of A325021 with respect to A001599.
%H A325022 Amiram Eldar, <a href="/A325022/b325022.txt">Table of n, a(n) for n = 1..682</a> (terms below 10^14)
%e A325022 140 is a term because 140*(140-tau(140))/sigma(140) = 140*(140-12)/336 = 160/3.
%t A325022 Select[Range[10^5], And[IntegerQ@ HarmonicMean@ #4, ! IntegerQ[#1 (#1 - #2)/#3]] & @@ Append[{#}~Join~DivisorSigma[{0, 1}, #], Divisors@ #] &] (* _Michael De Vlieger_, Mar 30 2019 *)
%o A325022 (Magma) [n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n) * n) / SumOfDivisors(n)) and not IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n))]
%o A325022 (PARI) isok(m) = my(d=numdiv(m), s=sigma(m)); !frac(m*d/s) && frac(m*(m-d)/s); \\ _Michel Marcus_, Mar 28 2019
%o A325022 (Python)
%o A325022 from itertools import count, islice
%o A325022 from math import prod
%o A325022 from functools import reduce
%o A325022 from sympy import factorint
%o A325022 def A325022_gen(startvalue=1): # generator of terms >= startvalue
%o A325022     for n in count(max(startvalue,1)):
%o A325022         f = factorint(n)
%o A325022         s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
%o A325022         if n*n%s and not reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s:
%o A325022             yield n
%o A325022 A325022_list = list(islice(A325022_gen(),10)) # _Chai Wah Wu_, Feb 14 2023
%Y A325022 Cf. A000005, A000203, A001599, A049820, A325020, A325021, A325023, A325024.
%K A325022 nonn
%O A325022 1,1
%A A325022 _Jaroslav Krizek_, Mar 28 2019