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A007357 Infinitary perfect numbers.
(Formerly M4267)
17
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)
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

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.

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

CROSSREFS

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

Sequence in context: A185288 A189000 A007358 * A002827 A137498 A036283

Adjacent sequences:  A007354 A007355 A007356 * A007358 A007359 A007360

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

More terms from Eric W. Weisstein, Jan 27 2004

STATUS

approved

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Last modified September 20 03:58 EDT 2014. Contains 246983 sequences.