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A054979
e-perfect numbers: numbers k such that the sum of the e-divisors (exponential divisors) of k equals 2*k.
39
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
1,1
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
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)
Brahim Mittou, Note on coreful divisor functions, Math. Montisnigri LXIV (2025) 57-63. See p. 58.
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.
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
KEYWORD
nonn
AUTHOR
Jud McCranie, May 29 2000
STATUS
approved