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A245782 Refactorable multiply-perfect numbers.

%I A245782 #25 Jan 13 2025 12:27:23
%S A245782 1,672,30240,23569920,45532800,14182439040,153003540480,403031236608,
%T A245782 518666803200,13661860101120,740344994887680,796928461056000,
%U A245782 212517062615531520,87934476737668055040,154345556085770649600,170206605192656148480,1161492388333469337600,1802582780370364661760
%N A245782 Refactorable multiply-perfect numbers.
%C A245782 Multiply-perfect numbers k (A007691) such that k / tau(k) is an integer.
%C A245782 Also multiply-perfect numbers k (A007691) such that (k / tau(k) - sigma(k) / k) = (k / A000005(k) - A000203(k) / k) is an integer.
%C A245782 Also multiply-perfect numbers k (A007691) such that (k / tau(k) + sigma(k) / k) = (k / A000005(k) + A000203(k) / k) is an integer.
%H A245782 Amiram Eldar, <a href="/A245782/b245782.txt">Table of n, a(n) for n = 1..407</a>
%e A245782 Multiply-perfect number 672 is in sequence because 672 / tau(672) = 28 (integer).
%t A245782 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 A245782 (Magma) [n:n in [A007691(n)] | (Denominator((n/(#[d: d in Divisors(n)]))-(SumOfDivisors(n)/n))) eq 1];
%o A245782 (PARI) isok(n) = !(n % numdiv(n)) && !(sigma(n) % n); \\ _Michel Marcus_, Aug 11 2014
%o A245782 (PARI) is(k) = {my(f = factor(k), s = sigma(f), d = numdiv(f)); !(s % k) && !(k % d);} \\ _Amiram Eldar_, May 09 2024
%Y A245782 Intersection of A033950 (refactorable numbers) and A007691 (multiply-perfect numbers).
%Y A245782 Subsequence of A245778 and A245786.
%Y A245782 Supersequence of A047728.
%Y A245782 Cf. A000005, A000203.
%Y A245782 Cf. A245776, A245777, A245779.
%K A245782 nonn
%O A245782 1,2
%A A245782 _Jaroslav Krizek_, Aug 01 2014
%E A245782 a(14)-a(18) from _Amiram Eldar_, May 09 2024