cp's OEIS Frontend

A328133 Exponential (2,4)-perfect numbers: numbers m such that esigma(esigma(m)) = 4m, where esigma(m) is the sum of exponential divisors of m (A051377).

%I A328133 #22 Jan 09 2025 10:36:44
%S A328133 540,3780,5940,7020,9180,10260,12420,15660,16740,19980,22140,23220,
%T A328133 25380,28620,31860,32940,36180,38340,39420,41580,42660,44820,48060,
%U A328133 49140,52380,54540,55620,57780,58860,61020,64260,68580,70740,71820,73980,75060,77220,80460
%N A328133 Exponential (2,4)-perfect numbers: numbers m such that esigma(esigma(m)) = 4m, where esigma(m) is the sum of exponential divisors of m (A051377).
%C A328133 Conjecturally, a subsequence of A083207 (tested for the first 659 terms of this sequence). - _Ivan N. Ianakiev_, Oct 05 2019
%D A328133 József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 1, p. 53.
%H A328133 Amiram Eldar, <a href="/A328133/b328133.txt">Table of n, a(n) for n = 1..10000</a>
%H A328133 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>.
%e A328133 540 is in the sequence since esigma(540) = 900, and esigma(900) = 2160 = 4*540.
%t A328133 f[p_, e_] := DivisorSum[e, p^# &]; esigma[1] = 1; esigma[n_] := Times @@ f @@@ FactorInteger[n]; espQ[n_] := esigma[esigma[n]] == 4n; Select[Range[80000], espQ]
%o A328133 (PARI) esigma(n) = {my(f = factor(n)); prod(k = 1, #f~, sumdiv(f[k, 2], d, f[k, 1]^d));}
%o A328133 isok(k) = esigma(esigma(k)) == 4*k; \\ _Amiram Eldar_, Jan 09 2025
%Y A328133 The exponential version of A019282.
%Y A328133 Cf. A051377, A083207.
%K A328133 nonn
%O A328133 1,1
%A A328133 _Amiram Eldar_, Oct 04 2019