A007357 Infinitary perfect numbers.

6, 60, 90, 36720, 12646368, 22276800, 126463680, 4201148160, 28770487200, 287704872000, 1446875426304, 2548696550400, 14468754263040, 590325173932032, 3291641594841600, 8854877608980480, 32916415948416000

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

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

Examples

			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
		

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.

Crossrefs

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

Programs

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 *)

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

Extensions

More terms from Eric W. Weisstein, Jan 27 2004