A038182 3-infinitary perfect numbers k: 3-i-sigma(k) = 2*k, where 3-i-sigma = A049418.

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

Offset: 1

Author

Yasutoshi Kohmoto

Keywords

• nonn
• nice
• more

Comments

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

Examples

			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.
		

Links

• J. O. M. Pedersen, Tables of Aliquot Cycles: backup on web.archive.org of no more existing web page, as of May 2014.
• J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]

Crossrefs

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

Programs

Mathematica

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 *)

PARI

is_A038182(n)=A049418(n)==2*n \\ M. F. Hasler, Sep 21 2022

Extensions

Definition shortened by R. J. Mathar, Oct 06 2010