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).

140, 270, 1638, 2970, 6200, 8190, 18600, 18620, 27846, 55860, 105664, 117800, 167400, 173600, 237510, 242060, 360360, 539400, 726180, 753480, 950976, 1089270, 1421280, 1539720, 2229500, 2290260, 2457000, 2845800, 4358600, 4713984, 4754880, 5772200, 6051500

Offset: 1

Author

Jaroslav Krizek, Mar 28 2019

Keywords

• nonn

Comments

Numbers m such that sigma(m) divides m*tau(m) but sigma(m) does not divide m*(m-tau(m)).

Complement of A325021 with respect to A001599.

Examples

			140 is a term because 140*(140-tau(140))/sigma(140) = 140*(140-12)/336 = 160/3.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..682 (terms below 10^14)

Crossrefs

Cf. A000005, A000203, A001599, A049820, A325020, A325021, A325023, A325024.

Programs

Magma

[n: n in [1..1000000] | IsIntegral((NumberOfDivisors(n) * n) / SumOfDivisors(n)) and not IsIntegral(((n-NumberOfDivisors(n)) * n) / SumOfDivisors(n))]

Mathematica

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 *)

PARI

isok(m) = my(d=numdiv(m), s=sigma(m)); !frac(m*d/s) && frac(m*(m-d)/s); \\ Michel Marcus, Mar 28 2019

Python

from itertools import count, islice
from math import prod
from functools import reduce
from sympy import factorint
def A325022_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 n*n%s and not reduce(lambda x,y:x*y%s,(e+1 for e in f.values()),1)*n%s:
            yield n
A325022_list = list(islice(A325022_gen(),10)) # Chai Wah Wu, Feb 14 2023