cp's OEIS Frontend

A325020 Numbers m 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 A325020 #24 May 10 2024 04:07:59
%S A325020 1,2,6,22,28,76,84,90,96,170,216,248,252,496,520,532,588,672,700,852,
%T A325020 864,1240,2176,2448,2480,2812,3360,6048,7392,7584,8128,9120,11480,
%U A325020 12616,12768,13832,14056,14720,15456,19488,20536,21216,27000,30240,31584,31968
%N A325020 Numbers m 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 A325020 Even perfect numbers from A000396 are terms.
%C A325020 Corresponding values of integers h: 0, 0, 1, 11, 11, 38, 27, 30, 32, 85, 72, 124, 81, ...
%C A325020 Supersequence of A325021 and A325023.
%H A325020 Amiram Eldar, <a href="/A325020/b325020.txt">Table of n, a(n) for n = 1..1835</a>
%e A325020 28 is a term because 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).
%t A325020 Select[Range[10^5], IntegerQ[#1 (#1 - #2)/#3] & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* _Michael De Vlieger_, Mar 24 2019 *)
%o A325020 (Magma) [n: n in [1..1000000] | IsIntegral((n - NumberOfDivisors(n)) * n / SumOfDivisors(n))]
%o A325020 (PARI) isok(m) = frac(m*(m-numdiv(m))/sigma(m)) == 0; \\ _Michel Marcus_, Mar 25 2019
%o A325020 (Python)
%o A325020 from itertools import count, islice
%o A325020 from math import prod
%o A325020 from functools import reduce
%o A325020 from sympy import factorint
%o A325020 def A325020_gen(startvalue=1): # generator of terms >= startvalue
%o A325020     for n in count(max(startvalue,1)):
%o A325020         f = factorint(n)
%o A325020         s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
%o A325020         if not (n-reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1))*n%s:
%o A325020             yield n
%o A325020 A325020_list = list(islice(A325020_gen(),20)) # _Chai Wah Wu_, Feb 14 2023
%Y A325020 Cf. A000005, A000203, A000396, A001599, A049820, A325021, A325022, A325023, A325024.
%K A325020 nonn
%O A325020 1,2
%A A325020 _Jaroslav Krizek_, Mar 24 2019