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A007357 Infinitary perfect numbers.

%I A007357 M4267 #51 Feb 16 2025 08:32:31
%S A007357 6,60,90,36720,12646368,22276800,126463680,4201148160,28770487200,
%T A007357 287704872000,1446875426304,2548696550400,14468754263040,
%U A007357 590325173932032,3291641594841600,8854877608980480,32916415948416000
%N A007357 Infinitary perfect numbers.
%C A007357 Numbers N whose sum of infinitary divisors equals 2*N: A049417(N)=2*N. - _Joerg Arndt_, Mar 20 2011
%C A007357 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
%D A007357 G. L. Cohen, personal communication.
%D A007357 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007357 G. L. Cohen, <a href="http://dx.doi.org/10.1090/S0025-5718-1990-0993927-5">On an integer's infinitary divisors</a>, Math. Comp., 54 (1990), 395-411.
%H A007357 A. V. Lelechenko, <a href="http://taac.org.ua/files/a2014/proceedings/UA-2-Andrew%20Lelechenko-440.pdf">The Quest for the Generalized Perfect Numbers</a>, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev.
%H A007357 David Moews, <a href="http://djm.cc/aliquot-database/aliquot-database-.1.txt">A database of aliquot cycles - Known infinitary perfect numbers (together with unitary perfect and e-perfect ones)</a>.
%H A007357 Jan Munch Pedersen, <a href="http://amicable.adsl.dk/aliquot/i1/i1.txt">Known infinitary perfect numbers</a>. [BROKEN LINK]
%H A007357 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InfinitaryPerfectNumber.html">Infinitary Perfect Number.</a>
%F A007357 {n: A049417(n) = 2*n}. - _R. J. Mathar_, Mar 18 2011
%F A007357 a(n)==0 (mod 6). Thus there are no odd infinitary perfect numbers. - _Vladimir Shevelev_, Mar 02 2011
%e A007357 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
%p A007357 isA007357 := proc(n)
%p A007357     A049417(n) = 2*n ;
%p A007357     simplify(%) ;
%p A007357 end proc:
%p A007357 for n from 1 do
%p A007357     if isA007357(n) then
%p A007357         printf("%d,\n",n) ;
%p A007357     end if;
%p A007357 end do: # _R. J. Mathar_, Oct 05 2017
%t A007357 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&])]]];
%t A007357 For[n = 6, True, n += 6, If[infiPerfQ[n], Print[n]]] (* _Jean-François Alcover_, Feb 08 2021 *)
%Y A007357 Cf. A129656 (infinitary abundant), A129657 (infinitary deficient).
%K A007357 nonn
%O A007357 1,1
%A A007357 _N. J. A. Sloane_
%E A007357 More terms from _Eric W. Weisstein_, Jan 27 2004