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%I #6 Jan 09 2020 18:12:49
%S 6,60,90,3024,133056,1330560,6879600,28828800,302702400,698544000,
%T 11763214848
%N Zeckendorf-infinitary perfect numbers: numbers k such that A331107(k) = 2*k.
%C No more terms below 4*10^10.
%e 6 is a term since A331107(6) = 12 = 2*6.
%t fb[n_] := Block[{k = Ceiling[Log[GoldenRatio, n*Sqrt[5]]], t = n, fr = {}}, While[k > 1, If[t >= Fibonacci[k], AppendTo[fr, 1]; t = t - Fibonacci[k], AppendTo[fr, 0]]; k--]; Fibonacci[1 + Position[Reverse@fr, _?(# == 1 &)]]]; f[p_, e_] := p^fb[e]; zsigma[1] = 1; zsigma[n_] := Times @@ (Flatten@(f @@@ FactorInteger[n]) + 1); zPerfectQ[n_] := zsigma[n] == 2 n; Select[Range[10^4], zPerfectQ] (* after _Robert G. Wilson v_ at A014417 *)
%Y Cf. A007357, A038182, A074849, A097464, A331107.
%K nonn,more
%O 1,1
%A _Amiram Eldar_, Jan 09 2020