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A054979 e-perfect numbers: numbers n such that the sum of the e-divisors (exponential divisors) of n equals 2n. 29
36, 180, 252, 396, 468, 612, 684, 828, 1044, 1116, 1260, 1332, 1476, 1548, 1692, 1800, 1908, 1980, 2124, 2196, 2340, 2412, 2556, 2628, 2700, 2772, 2844, 2988, 3060, 3204, 3276, 3420, 3492, 3636, 3708, 3852, 3924, 4068, 4140, 4284, 4572, 4716 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The e-divisors (or exponential divisors) of x=Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.

The number of e-divisors for n is A049419(n). - Jon Perry, Nov 13 2012

Conjecture: Every e-perfect number is divisible by 36, see A219016. - Jon Perry, Nov 13 2012

REFERENCES

R. K. Guy, Unsolved Problems In Number Theory, B17.

LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Eric Weisstein's World of Mathematics, e-Perfect Number

FORMULA

{n: A051377(n) = 2*n}. - R. J. Mathar, Oct 05 2017

EXAMPLE

The e-divisors of 36 are 2*3, 4*3, 2*9 and 4*9 and the sum of these = 2*36, so 36 is e-perfect.

MAPLE

for n from 1 do

    if A051377(n) = 2*n then

        printf("%d, \n", n) ;

    end if;

end do: # R. J. Mathar, Oct 05 2017

MATHEMATICA

ee[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; Select[Range[5000], ee[#] == 2 # &] (* T. D. Noe, Nov 14 2012 *)

PROG

(PARI) is(n)=my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d))==2*n \\ Charles R Greathouse IV, Nov 22 2011

CROSSREFS

Cf. A051377, A054980, A219016.

Sequence in context: A318100 A335218 A321145 * A335219 A102949 A211733

Adjacent sequences:  A054976 A054977 A054978 * A054980 A054981 A054982

KEYWORD

nonn

AUTHOR

Jud McCranie, May 29 2000

STATUS

approved

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Last modified August 18 02:34 EDT 2022. Contains 356204 sequences. (Running on oeis4.)