

A007357


Infinitary perfect numbers.
(Formerly M4267)


30



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



FORMULA

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)
simplify(%) ;
end proc:
for n from 1 do
if isA007357(n) then
printf("%d, \n", n) ;
end if;


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&])]]];


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



