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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.

%I A054979 #45 Feb 16 2025 08:32:42
%S A054979 36,180,252,396,468,612,684,828,1044,1116,1260,1332,1476,1548,1692,
%T A054979 1800,1908,1980,2124,2196,2340,2412,2556,2628,2700,2772,2844,2988,
%U A054979 3060,3204,3276,3420,3492,3636,3708,3852,3924,4068,4140,4284,4572,4716
%N A054979 e-perfect numbers: numbers k such that the sum of the e-divisors (exponential divisors) of k equals 2*k.
%C A054979 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.
%C A054979 The number of e-divisors for n is A049419(n). - _Jon Perry_, Nov 13 2012
%C A054979 Conjecture: Every e-perfect number is divisible by 36, see A219016. - _Jon Perry_, Nov 13 2012
%D A054979 Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B17, pp. 110-111.
%D A054979 József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 116-117.
%H A054979 Amiram Eldar, <a href="/A054979/b054979.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H A054979 E. G. Straus and M. V. Subbarao, <a href="https://doi.org/10.1215/S0012-7094-74-04152-0">On exponential divisors</a>, Duke Math. J., Vol. 41 (1974), pp. 465-471.
%H A054979 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e-PerfectNumber.html">e-Perfect Number</a>.
%F A054979 {n: A051377(n) = 2*n}. - _R. J. Mathar_, Oct 05 2017
%e A054979 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.
%p A054979 for n from 1 do
%p A054979     if A051377(n) = 2*n then
%p A054979         printf("%d,\n",n) ;
%p A054979     end if;
%p A054979 end do: # _R. J. Mathar_, Oct 05 2017
%t A054979 ee[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; Select[Range[5000], ee[#] == 2 # &] (* _T. D. Noe_, Nov 14 2012 *)
%o A054979 (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
%Y A054979 Cf. A051377, A054980, A219016.
%K A054979 nonn
%O A054979 1,1
%A A054979 _Jud McCranie_, May 29 2000