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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).
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%I #24 May 10 2024 04:07:59

%S 1,2,6,22,28,76,84,90,96,170,216,248,252,496,520,532,588,672,700,852,

%T 864,1240,2176,2448,2480,2812,3360,6048,7392,7584,8128,9120,11480,

%U 12616,12768,13832,14056,14720,15456,19488,20536,21216,27000,30240,31584,31968

%N 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 Even perfect numbers from A000396 are terms.

%C Corresponding values of integers h: 0, 0, 1, 11, 11, 38, 27, 30, 32, 85, 72, 124, 81, ...

%C Supersequence of A325021 and A325023.

%H Amiram Eldar, <a href="/A325020/b325020.txt">Table of n, a(n) for n = 1..1835</a>

%e 28 is a term because 28*(28-tau(28))/sigma(28) = 28*(28-6)/56 = 11 (integer).

%t Select[Range[10^5], IntegerQ[#1 (#1 - #2)/#3] & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* _Michael De Vlieger_, Mar 24 2019 *)

%o (Magma) [n: n in [1..1000000] | IsIntegral((n - NumberOfDivisors(n)) * n / SumOfDivisors(n))]

%o (PARI) isok(m) = frac(m*(m-numdiv(m))/sigma(m)) == 0; \\ _Michel Marcus_, Mar 25 2019

%o (Python)

%o from itertools import count, islice

%o from math import prod

%o from functools import reduce

%o from sympy import factorint

%o def A325020_gen(startvalue=1): # generator of terms >= startvalue

%o for n in count(max(startvalue,1)):

%o f = factorint(n)

%o s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())

%o if not (n-reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1))*n%s:

%o yield n

%o A325020_list = list(islice(A325020_gen(),20)) # _Chai Wah Wu_, Feb 14 2023

%Y Cf. A000005, A000203, A000396, A001599, A049820, A325021, A325022, A325023, A325024.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Mar 24 2019