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
G. L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.
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 e-perfect 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]]] (* Jean-François Alcover, Feb 08 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Eric W. Weisstein, Jan 27 2004
STATUS
approved