A054979 e-perfect numbers: numbers k such that the sum of the e-divisors (exponential divisors) of k equals 2*k.

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

Offset: 1

Author

Jud McCranie, May 29 2000

Keywords

• nonn

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

Examples

			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.
		

References

• Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B17, pp. 110-111.
• József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 116-117.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Math. J., Vol. 41 (1974), pp. 465-471.
• Eric Weisstein's World of Mathematics, e-Perfect Number.

Crossrefs

Cf. A051377, A054980, A219016.

Programs

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 *)

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

Formula

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