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3-infinitary perfect numbers k: 3-i-sigma(k) = 2*k, where 3-i-sigma = A049418.
6

%I #20 Oct 24 2024 09:25:48

%S 6,28,3024,6552,27578880,49266240,49095705098695680

%N 3-infinitary perfect numbers k: 3-i-sigma(k) = 2*k, where 3-i-sigma = A049418.

%C Similarly, we have 3-i-sigma(x)/x = r for the following numbers: r = 3 for x = 672, 13104, 4021920, 55157760, 98532480, 459818240, 372667889664, 7267023848448, 1178296922368696320, 5498718971053916160, ...; r = 4 for x = 2178540; r = 3/2 for x = 2, 24, 9192960, 196382820394782720. (Values above 10^7 from _Yasutoshi Kohmoto_, some terms may be missing.) - _M. F. Hasler_, Sep 21 2022

%H J. O. M. Pedersen, <a href="http://web.archive.org/web/20140502102524/http://amicable.homepage.dk/tables.htm">Tables of Aliquot Cycles</a>: backup on web.archive.org of no more existing web page, as of May 2014.

%H J. O. M. Pedersen, <a href="/A063990/a063990.pdf">Tables of Aliquot Cycles</a>. [Cached copy, pdf file only]

%e Factorizations: 2*3, 2^2*7, 2^4*3^3*7, 2^3*3^2*7*13, 2^9*3^4*5*7*19, 2^6*3*5*19*37*73, 2^10*3^6*5*19^2*127*379*757.

%t f[p_, e_] := Module[{d = IntegerDigits[e, 3]}, m = Length[d]; Product[(p^((d[[j]] + 1)*3^(m - j)) - 1)/(p^(3^(m - j)) - 1), {j, 1, m}]]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[7000], s[#] == 2*# &] (* _Amiram Eldar_, Oct 24 2024 *)

%o (PARI) is_A038182(n)=A049418(n)==2*n \\ _M. F. Hasler_, Sep 21 2022

%Y Cf. A007357, A037445, A038148, A049418, A074849, A097464.

%K nonn,nice,more

%O 1,1

%A _Yasutoshi Kohmoto_

%E Definition shortened by _R. J. Mathar_, Oct 06 2010