OFFSET
1,1
COMMENTS
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.
a(10) > 10^15. - Donovan Johnson, Nov 22 2011
The following numbers also belong to this sequence; however, their actual positions are unknown: 168136940595306022660197936246988800, 11712310558743727210993873194516480000, 1307484087615221689700651798824550400000. - Andrew Lelechenko, Apr 01 2014
The number of terms with a given number of distinct prime divisors is finite (Straus and Subbarao, 1974). - Amiram Eldar, Mar 04 2021
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
Andrew V. Lelechenko, Exponential and infinitary divisors, Ukrainian Mathematical Journal, Vol. 68, No. 8 (2017), pp. 1222-1237; arXiv preprint, arXiv:1405.7597 [math.NT], 2014.
Jan Munch Pedersen, Exponential Perfect Numbers.
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.
EXAMPLE
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).
PROG
(PARI) eperfect(n)=my(f=factor(n)); prod(i=1, #f[, 1], sumdiv(f[i, 2], d, f[i, 1]^d))==2*n
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
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Jud McCranie, May 29 2000
EXTENSIONS
a(9) from Donovan Johnson, Nov 22 2011
STATUS
approved