%I #50 Mar 04 2021 07:02:57

%S 36,1800,2700,17424,1306800,4769856,238492800,357739200,54531590400

%N Primitive e-perfect numbers: primitive elements of the e-perfect numbers (A054979).

%C The nonprimitive e-perfect numbers are obtained from the primitive ones by multiplying by m, if m is squarefree and relatively prime to the primitive e-perfect number.

%C a(10) > 10^15. - _Donovan Johnson_, Nov 22 2011

%C The following numbers also belong to this sequence; however, their actual positions are unknown: 168136940595306022660197936246988800, 11712310558743727210993873194516480000, 1307484087615221689700651798824550400000. - _Andrew Lelechenko_, Apr 01 2014

%C The number of terms with a given number of distinct prime divisors is finite (Straus and Subbarao, 1974). - _Amiram Eldar_, Mar 04 2021

%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B17, pp. 110-111.

%D József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, Chapter III, p. 116-117.

%H Andrew V. Lelechenko, <a href="https://doi.org/10.1007/s11253-017-1289-7">Exponential and infinitary divisors</a>, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; <a href="http://arxiv.org/abs/1405.7597">arXiv preprint</a>, arXiv:1405.7597 [math.NT], 2014.

%H Jan Munch Pedersen, <a href="http://62.198.248.44/aliquot/e1/e1.txt">Exponential Perfect Numbers</a>.

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/e-PerfectNumber.html">e-Perfect Number</a>.

%e 180 = 36*5 (nonprimitive). 252 = 36*7 (nonprimitive). 1260 = 36*5*7 (nonprimitive). 1800 = 36*5^2 (primitive, 5^2 not squarefree and coprime to 36).

%o (PARI) eperfect(n)=my(f=factor(n));prod(i=1,#f[,1],sumdiv(f[i,2],d, f[i,1]^d))==2*n

%o is(n)=if(!eperfect(n),0,my(f=factor(n));for(i=1,#f[,1],if(f[i,2]==1&&eperfect(n/f[i,1]),return(0)));1) \\ _Charles R Greathouse IV_, Nov 22 2011

%Y Cf. A051377, A054979, A160134 (complement).

%K nonn,more

%O 1,1

%A _Jud McCranie_, May 29 2000

%E a(9) from _Donovan Johnson_, Nov 22 2011