OFFSET
1,2
COMMENTS
All primes except 2 and 3, and squarefree odd numbers, are in this sequence. [It appears that the author may have meant squarefree numbers coprime to 6, not squarefree odd numbers. - Peter Munn, Nov 22 2020]
From Amiram Eldar, Dec 05 2025: (Start)
It was not proved that all e-perfect numbers are divisible by 36.
Straus and Subbarao (1974) proved that there are no odd e-perfect numbers, and therefore all the e-perfect number are divisible by 4. But the question of the existence of an e-perfect number that is not divisible by 3 is still open.
Fabrykowski and Subbarao (1986) proved that if there is an e-perfect number that is not divisible by 3, then it is larger than 10^664, has at least 118 distinct prime divisors, and is divisible by 2^117.
If all the e-perfect numbers are divisible by 36, then the asymptotic density of this sequence is 36 * A318645 = 0.31299098... . (End)
REFERENCES
J. Fabrykowski and M. V. Subbarao, On e-perfect numbers not divisible by 3, Nieuw Arch. Wiskunde (4), Vol. 4 (1986), pp. 165-173.
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Math. J., Vol. 41, No. 2 (1974), pp. 465-471.
EXAMPLE
A054979(3)=252 so a(3)=7.
The e-divisors of 36*50 = 1800 are 30, 120, 90, 360, 150, 600, 450 and 1800, which sum to 3600 as required.
MATHEMATICA
ee[n_] := Times @@ (Sum[First[#]^d, {d, Divisors[Last[#]]}] &) /@ FactorInteger[n]; Select[Range[10000], ee[#] == 2 # &]/36 (* T. D. Noe, Nov 14 2012 *)
PROG
(PARI) is(n)=my(f=factor(36*n)); prod(i=1, #f~, sumdiv(f[i, 2], d, f[i, 1]^d))==72*n \\ Charles R Greathouse IV, Dec 30 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Jon Perry, Nov 13 2012
STATUS
approved