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).

1, 2, 6, 22, 28, 76, 84, 90, 96, 170, 216, 248, 252, 496, 520, 532, 588, 672, 700, 852, 864, 1240, 2176, 2448, 2480, 2812, 3360, 6048, 7392, 7584, 8128, 9120, 11480, 12616, 12768, 13832, 14056, 14720, 15456, 19488, 20536, 21216, 27000, 30240, 31584, 31968

Offset: 1

Author

Jaroslav Krizek, Mar 24 2019

Keywords

• nonn

Comments

Even perfect numbers from A000396 are terms.

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

Supersequence of A325021 and A325023.

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..1835

Crossrefs

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

Programs

Magma

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

Mathematica

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

PARI

isok(m) = frac(m*(m-numdiv(m))/sigma(m)) == 0; \\ Michel Marcus, Mar 25 2019

Python

from itertools import count, islice
from math import prod
from functools import reduce
from sympy import factorint
def A325020_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        s = prod((p**(e+1)-1)//(p-1) for p, e in f.items())
        if not (n-reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1))*n%s:
            yield n
A325020_list = list(islice(A325020_gen(),20)) # Chai Wah Wu, Feb 14 2023