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A007357 Infinitary perfect numbers.
(Formerly M4267)
30

%I M4267 #50 Dec 26 2021 21:39:09

%S 6,60,90,36720,12646368,22276800,126463680,4201148160,28770487200,

%T 287704872000,1446875426304,2548696550400,14468754263040,

%U 590325173932032,3291641594841600,8854877608980480,32916415948416000

%N Infinitary perfect numbers.

%C Numbers N whose sum of infinitary divisors equals 2*N: A049417(N)=2*N. - _Joerg Arndt_, Mar 20 2011

%C 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 G. L. Cohen, personal communication.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H 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 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 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 Jan Munch Pedersen, <a href="http://amicable.adsl.dk/aliquot/i1/i1.txt">Known infinitary perfect numbers</a>. [BROKEN LINK]

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InfinitaryPerfectNumber.html">Infinitary Perfect Number.</a>

%F {n: A049417(n) = 2*n}. - _R. J. Mathar_, Mar 18 2011

%F a(n)==0 (mod 6). Thus there are no odd infinitary perfect numbers. - _Vladimir Shevelev_, Mar 02 2011

%e 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 isA007357 := proc(n)

%p A049417(n) = 2*n ;

%p simplify(%) ;

%p end proc:

%p for n from 1 do

%p if isA007357(n) then

%p printf("%d,\n",n) ;

%p end if;

%p end do: # _R. J. Mathar_, Oct 05 2017

%t 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 For[n = 6, True, n += 6, If[infiPerfQ[n], Print[n]]] (* _Jean-François Alcover_, Feb 08 2021 *)

%Y Cf. A129656 (infinitary abundant), A129657 (infinitary deficient).

%K nonn

%O 1,1

%A _N. J. A. Sloane_

%E More terms from _Eric W. Weisstein_, Jan 27 2004

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)