A245782 Refactorable multiply-perfect numbers.

1, 672, 30240, 23569920, 45532800, 14182439040, 153003540480, 403031236608, 518666803200, 13661860101120, 740344994887680, 796928461056000, 212517062615531520, 87934476737668055040, 154345556085770649600, 170206605192656148480, 1161492388333469337600, 1802582780370364661760

Offset: 1

Author

Jaroslav Krizek, Aug 01 2014

Keywords

• nonn

Comments

Multiply-perfect numbers k (A007691) such that k / tau(k) is an integer.

Also multiply-perfect numbers k (A007691) such that (k / tau(k) - sigma(k) / k) = (k / A000005(k) - A000203(k) / k) is an integer.

Also multiply-perfect numbers k (A007691) such that (k / tau(k) + sigma(k) / k) = (k / A000005(k) + A000203(k) / k) is an integer.

Examples

			Multiply-perfect number 672 is in sequence because 672 / tau(672) = 28 (integer).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..407

Crossrefs

Intersection of A033950 (refactorable numbers) and A007691 (multiply-perfect numbers).

Subsequence of A245778 and A245786.

Supersequence of A047728.

Cf. A000005, A000203.

Cf. A245776, A245777, A245779.

Programs

Magma

[n:n in [A007691(n)] | (Denominator((n/(#[d: d in Divisors(n)]))-(SumOfDivisors(n)/n))) eq 1];

Mathematica

q[n_] := Module[{d = DivisorSigma[0, n], s = DivisorSigma[1, n]}, Divisible[s, n] && Divisible[n, d]]; Select[Range[31000], q] (* Amiram Eldar, May 09 2024 *)

PARI

isok(n) = !(n % numdiv(n)) && !(sigma(n) % n); \\ Michel Marcus, Aug 11 2014

PARI

is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && !(k % d);} \\ Amiram Eldar, May 09 2024

Extensions

a(14)-a(18) from Amiram Eldar, May 09 2024