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A007357
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Infinitary perfect numbers.
(Formerly M4267)
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30
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6, 60, 90, 36720, 12646368, 22276800, 126463680, 4201148160, 28770487200, 287704872000, 1446875426304, 2548696550400, 14468754263040, 590325173932032, 3291641594841600, 8854877608980480, 32916415948416000
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OFFSET
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1,1
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COMMENTS
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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
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REFERENCES
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G. L. Cohen, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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a(n)==0 (mod 6). Thus there are no odd infinitary perfect numbers. - Vladimir Shevelev, Mar 02 2011
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EXAMPLE
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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
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MAPLE
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isA007357 := proc(n)
simplify(%) ;
end proc:
for n from 1 do
if isA007357(n) then
printf("%d, \n", n) ;
end if;
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MATHEMATICA
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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&])]]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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