A245782 - OEIS

%I #25 Jan 13 2025 12:27:23

%S 1,672,30240,23569920,45532800,14182439040,153003540480,403031236608,

%T 518666803200,13661860101120,740344994887680,796928461056000,

%U 212517062615531520,87934476737668055040,154345556085770649600,170206605192656148480,1161492388333469337600,1802582780370364661760

%N Refactorable multiply-perfect numbers.

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

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

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

%H Amiram Eldar, <a href="/A245782/b245782.txt">Table of n, a(n) for n = 1..407</a>

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

%t 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 *)

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

%o (PARI) isok(n) = !(n % numdiv(n)) && !(sigma(n) % n); \\ _Michel Marcus_, Aug 11 2014

%o (PARI) is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && !(k % d);} \\ _Amiram Eldar_, May 09 2024

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

%Y Subsequence of A245778 and A245786.

%Y Supersequence of A047728.

%Y Cf. A000005, A000203.

%Y Cf. A245776, A245777, A245779.

%K nonn

%O 1,2

%A _Jaroslav Krizek_, Aug 01 2014

%E a(14)-a(18) from _Amiram Eldar_, May 09 2024