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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).

%I A328120 #23 Jan 09 2025 10:36:02
%S A328120 9,12,45,60,63,84,99,117,132,153,156,171,204,207,228,261,270,276,279,
%T A328120 315,333,348,369,372,387,420,423,444,477,492,495,516,531,549,564,585,
%U A328120 603,636,639,657,660,693,708,711,732,747,765,780,801,804,819,852,855,873
%N 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).
%C A328120 Hanumanthachari et al. proved that:
%C A328120 1) The only e-superperfect number of the form p^q with p and q primes is 9 = 3^2.
%C A328120 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.
%C A328120 3) If k is squarefree coprime to esigma(m) then m*k is e-superperfect if and only if m is e-superperfect.
%D A328120 József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 53.
%H A328120 Amiram Eldar, <a href="/A328120/b328120.txt">Table of n, a(n) for n = 1..10000</a>
%H A328120 J. Hanumanthachari, V. V. Subrahmanya Sastri, and V. Srinivasan, On e-perfect numbers, Math. Student, Vol. 46, No. 1 (1978), pp. 71-80; <a href="https://schoolbooksarchive.azimpremjiuniversity.edu.in/handle/20.500.12497/11737">entire issue</a>.
%F A328120 9 is in the sequence since esigma(9) = 12 and esigma(12) = 18 = 2*9.
%t A328120 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]
%o A328120 (PARI) esigma(n) = {my(f = factor(n)); prod(k = 1, #f~, sumdiv(f[k, 2], d, f[k, 1]^d));}
%o A328120 isok(k) = esigma(esigma(k)) == 2*k; \\ _Amiram Eldar_, Jan 09 2025
%Y A328120 The exponential version of A019279.
%Y A328120 Cf. A038843, A051377.
%K A328120 nonn
%O A328120 1,1
%A A328120 _Amiram Eldar_, Oct 04 2019