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A328132 Exponential (2,3)-perfect numbers: numbers m such that esigma(esigma(m)) = 3m, where esigma(m) is the sum of exponential divisors of m (A051377).

%I A328132 #12 Jan 09 2025 09:37:54
%S A328132 300,2100,3300,3900,5100,5700,6900,8700,9300,11100,12100,12300,12900,
%T A328132 14100,15900,17700,18300,20100,21300,21900,23100,23700,23760,24900,
%U A328132 26700,27300,29100,30300,30900,32100,32700,33900,35700,38100,39300,39900,41100,41700,42900
%N A328132 Exponential (2,3)-perfect numbers: numbers m such that esigma(esigma(m)) = 3m, where esigma(m) is the sum of exponential divisors of m (A051377).
%D A328132 József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 53.
%H A328132 Amiram Eldar, <a href="/A328132/b328132.txt">Table of n, a(n) for n = 1..10000</a>
%H A328132 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 A328132 300 is in the sequence since esigma(300) = 540, and esigma(540) = 900 = 3*300.
%t A328132 f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; espQ[n_] := esigma[esigma[n]] == 3n; Select[Range[50000], espQ]
%o A328132 (PARI) esigma(n) = {my(f = factor(n)); prod(k = 1, #f~, sumdiv(f[k, 2], d, f[k, 1]^d));}
%o A328132 isok(k) = esigma(esigma(k)) == 3*k; \\ _Amiram Eldar_, Jan 09 2025
%Y A328132 The exponential version of A019281.
%Y A328132 Cf. A051377, A064012.
%K A328132 nonn
%O A328132 1,1
%A A328132 _Amiram Eldar_, Oct 04 2019