OFFSET
1,1
COMMENTS
The exponential version of A019279.
Hanumanthachari et al. proved that:
1) The only e-superperfect number of the form p^q with p and q primes is 9 = 3^2.
2) If p prime, m squarefree coprime to m with gcd(p+1, m) > 1 then p^2 * m is e-superperfect only if p = 2.
3) If k is squarefree coprime to esigma(m) then m*k is e-superperfect if and only if m is e-superperfect.
REFERENCES
J. Hanumanthachari, V. V. Subrahmanya Sastri, and V. Srinivasan, On e-perfect numbers, Math. Student, Vol. 46, No. 1 (1978), pp. 71-80.
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 53.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
9 is in the sequence since esigma(9) = 12 and esigma(12) = 18 = 2*9.
MATHEMATICA
f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; espQ[n_] := esigma[esigma[n]] == 2n; Select[Range[1000], espQ]
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Oct 04 2019
STATUS
approved