A328120 Exponential superperfect numbers (or e-superperfect numbers): numbers m such that esigma(esigma(m)) = 2m, where esigma(m) is the sum of exponential divisors of m (A051377).

9, 12, 45, 60, 63, 84, 99, 117, 132, 153, 156, 171, 204, 207, 228, 261, 270, 276, 279, 315, 333, 348, 369, 372, 387, 420, 423, 444, 477, 492, 495, 516, 531, 549, 564, 585, 603, 636, 639, 657, 660, 693, 708, 711, 732, 747, 765, 780, 801, 804, 819, 852, 855, 873

Offset: 1

Author

Amiram Eldar, Oct 04 2019

Keywords

• nonn

Comments

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

• 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
• J. Hanumanthachari, V. V. Subrahmanya Sastri, and V. Srinivasan, On e-perfect numbers, Math. Student, Vol. 46, No. 1 (1978), pp. 71-80; entire issue.

Crossrefs

The exponential version of A019279.

Cf. A038843, A051377.

Programs

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]

PARI

esigma(n) = {my(f = factor(n)); prod(k = 1, #f~, sumdiv(f[k, 2], d, f[k, 1]^d));}
isok(k) = esigma(esigma(k)) == 2*k; \\ Amiram Eldar, Jan 09 2025

Formula

9 is in the sequence since esigma(9) = 12 and esigma(12) = 18 = 2*9.