

A007357


Infinitary perfect numbers.
(Formerly M4267)


26



6, 60, 90, 36720, 12646368, 22276800, 126463680, 4201148160, 28770487200, 287704872000, 1446875426304, 2548696550400, 14468754263040, 590325173932032, 3291641594841600, 8854877608980480, 32916415948416000
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OFFSET

1,1


COMMENTS

Numbers N whose sum of infinitary divisors equals 2*N: A049417(N)=2*N.  Joerg Arndt, Mar 20 2011
6 is the only infinitary perfect number which is also perfect number (A000396). 6 is also the only squarefree infinitary perfect number.  Vladimir Shevelev, Mar 02 2011


REFERENCES

G. L. Cohen, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

Table of n, a(n) for n=1..17.
G. L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395411.
A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
David Moews, A database of aliquot cycles  Known infinitary perfect numbers (together with unitary perfect and eperfect ones).
Jan Munch Pedersen, Known infinitary perfect numbers. [BROKEN LINK]
Eric Weisstein's World of Mathematics, Infinitary Perfect Number.


FORMULA

{n: A049417(n) = 2*n}.  R. J. Mathar, Mar 18 2011
a(n)==0 (mod 6). Thus there are no odd infinitary perfect numbers.  Vladimir Shevelev, Mar 02 2011


EXAMPLE

Let n=90. Its unique expansion over distinct terms of A050376 is 90=2*5*9. Thus the infinitary divisors of 90 are 1,2,5,9,10,18,45,90. The number of such divisors is 2^3, i.e., the cardinality of the set of all subsets of the set {2,5,9}. The sum of such divisors is (2+1)*(5+1)*(9+1)=180 and the sum of proper such divisors is 90=n. Thus 90 is in the sequence.  Vladimir Shevelev, Mar 02 2011


MAPLE

isA007357 := proc(n)
A049417(n) = 2*n ;
simplify(%) ;
end proc:
for n from 1 do
if isA007357(n) then
printf("%d, \n", n) ;
end if;
end do: # R. J. Mathar, Oct 05 2017


MATHEMATICA

infiPerfQ[n_] := 2n == Total[If[n == 1, 1, Sort @ Flatten @ Outer[ Times, Sequence @@ (FactorInteger[n] /. {p_, m_Integer} :> p^Select[Range[0, m], BitOr[m, #] == m&])]]];
For[n = 6, True, n += 6, If[infiPerfQ[n], Print[n]]] (* JeanFrançois Alcover, Feb 08 2021 *)


CROSSREFS

Cf. A129656 (infinitary abundant), A129657 (infinitary deficient).
Sequence in context: A323757 A331108 A324707 * A327158 A002827 A331111
Adjacent sequences: A007354 A007355 A007356 * A007358 A007359 A007360


KEYWORD

nonn


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from Eric W. Weisstein, Jan 27 2004


STATUS

approved



