A325024 Multiply-perfect numbers m from A007691 such that m*(m-tau(m))/sigma(m) is not an integer where k-tau(k) is the number of the non-divisors of k (A049820) and sigma(k) is the sum of the divisors of k (A000203).

120, 523776, 459818240, 1476304896, 31998395520, 51001180160, 518666803200, 30823866178560, 740344994887680, 796928461056000, 212517062615531520, 69357059049509038080, 87934476737668055040, 170206605192656148480, 1161492388333469337600, 1802582780370364661760

Offset: 1

Author

Jaroslav Krizek, May 12 2019

Keywords

• nonn

Comments

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

Complement of A325023 with respect to A007691.

Examples

			120 is a term because 120*(120-tau(120))/sigma(120) = 120*(120-16)/360 = 104/3.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..218

Crossrefs

Cf. A000005, A000203, A007691, A049820, A325020, A325021, A325022, A325023.

Programs

Magma

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

Mathematica

Select[Range[10^6], And[Mod[#3, #1] == 0, !IntegerQ[#1 (#1 - #2)/#3]] & @@ Prepend[DivisorSigma[{0, 1}, #], #] &] (* Amiram Eldar, Jul 10 2019 after Michael De Vlieger at A325023 *)

PARI

isA325024(m) = { my(s=sigma(m)); ((1==denominator(s/m)) && (1!=denominator(m*(m-numdiv(m))/s))); }; \\ Antti Karttunen, May 25 2019