A331724 Multiply-perfect numbers (A007691) that are arithmetic (A003601).

1, 6, 672, 30240, 32760, 23569920, 45532800, 14182439040, 51001180160, 153003540480, 403031236608, 518666803200, 13661860101120, 740344994887680, 796928461056000, 212517062615531520, 87934476737668055040, 154345556085770649600, 170206605192656148480

Offset: 1

Author

Jaroslav Krizek, Jan 25 2020

Keywords

• nonn

Comments

Multiply-perfect numbers m such that values A(m) = sigma(m)/tau(m) = A000203(m)/A000005(m) are any integers.

Corresponding values of A(m): 1, 3, 84, 1260, 1365, 294624, 474300, 36933435, 318757376, 637514752, 1199497728, ...

Complement of A330533 with respect to A007691. Supersequence of A046985.

Has many terms in common with B = {multiply perfect numbers n divisible by bigomega(n)}: only {1, 45532800, 403031236608, 212517062615531520, ...} are in {a(n)} \ B, while {120, 523776, 2178540, ...} are in B \ {a(n)}. - M. F. Hasler, Jan 31 2020

Examples

			sigma(672)/tau(672) = 2016/24 = 84 (integers).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..423

Crossrefs

Cf. A000005, A000203, A003601, A007691, A046985, A330533.

Cf. A325025 (multiply-perfect numbers that are harmonic).

Programs

Magma

[m: m in [1..10^7] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and IsIntegral(SumOfDivisors(m) / m)];

Mathematica

seqQ[n_] := And @@ (Divisible[DivisorSigma[1, n], #] & /@ {n, DivisorSigma[0, n]}); Select[Range[5*10^7], seqQ] (* Amiram Eldar, Jan 25 2020 *)

PARI

is_A331724(n)={my(f=factor(n),s=sigma(f));!(s%n||s%numdiv(f))} \\ M. F. Hasler, Jan 31 2020