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

%I A331724 #25 Apr 10 2025 14:57:20
%S A331724 1,6,672,30240,32760,23569920,45532800,14182439040,51001180160,
%T A331724 153003540480,403031236608,518666803200,13661860101120,
%U A331724 740344994887680,796928461056000,212517062615531520,87934476737668055040,154345556085770649600,170206605192656148480
%N A331724 Multiply-perfect numbers (A007691) that are arithmetic (A003601).
%C A331724 Multiply-perfect numbers m such that values A(m) = sigma(m)/tau(m) = A000203(m)/A000005(m) are any integers.
%C A331724 Corresponding values of A(m): 1, 3, 84, 1260, 1365, 294624, 474300, 36933435, 318757376, 637514752, 1199497728, ...
%C A331724 Complement of A330533 with respect to A007691. Supersequence of A046985.
%C A331724 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
%H A331724 Amiram Eldar, <a href="/A331724/b331724.txt">Table of n, a(n) for n = 1..423</a>
%e A331724 sigma(672)/tau(672) = 2016/24 = 84 (integers).
%t A331724 seqQ[n_] := And @@ (Divisible[DivisorSigma[1, n], #] & /@ {n, DivisorSigma[0, n]}); Select[Range[5*10^7], seqQ] (* _Amiram Eldar_, Jan 25 2020 *)
%o A331724 (Magma) [m: m in [1..10^7] | IsIntegral(SumOfDivisors(m) / NumberOfDivisors(m)) and IsIntegral(SumOfDivisors(m) / m)];
%o A331724 (PARI) is_A331724(n)={my(f=factor(n),s=sigma(f));!(s%n||s%numdiv(f))} \\ _M. F. Hasler_, Jan 31 2020
%Y A331724 Cf. A000005, A000203, A003601, A007691, A046985, A330533.
%Y A331724 Cf. A325025 (multiply-perfect numbers that are harmonic).
%K A331724 nonn
%O A331724 1,2
%A A331724 _Jaroslav Krizek_, Jan 25 2020