A054980 Primitive e-perfect numbers: primitive elements of the e-perfect numbers (A054979).

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

Offset: 1

Author

Jud McCranie, May 29 2000

Keywords

• nonn
• more

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

Examples

			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).
		

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.

Crossrefs

Cf. A051377, A054979, A160134 (complement).

Programs

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

Extensions

a(9) from Donovan Johnson, Nov 22 2011