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A007357 Infinitary perfect numbers.
(Formerly M4267)
6, 60, 90, 36720, 12646368, 22276800, 126463680, 4201148160, 28770487200, 287704872000, 1446875426304, 2548696550400, 14468754263040, 590325173932032, 3291641594841600, 8854877608980480, 32916415948416000 (list; graph; refs; listen; history; text; internal format)



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


G. L. Cohen, personal communication.

A. V. Lelechenko, The Quest for the Generalized Perfect Numbers, in Theoretical and Applied Aspects of Cybernetics, TAAC 2014, Kiev; http://taac.org.ua/files/a2014/proceedings/UA-2-Andrew%20Lelechenko-440.pdf

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


Table of n, a(n) for n=1..17.

G. L. Cohen, On an integer's infinitary divisors, Math. Comp., 54 (1990), 395-411.

Jan Munch Pedersen, Known infinitary perfect numbers.

Eric Weisstein's World of Mathematics, Infinitary Perfect Number.


{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


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


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

Sequence in context: A185288 A189000 A007358 * A002827 A137498 A250070

Adjacent sequences:  A007354 A007355 A007356 * A007358 A007359 A007360




N. J. A. Sloane.


More terms from Eric W. Weisstein, Jan 27 2004



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Last modified February 10 20:58 EST 2016. Contains 268157 sequences.