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%I #32 Dec 27 2018 06:10:51
%S 6,28,60,84,90,120,336,496,840,924,1008,1080,1260,1320,1440,1680,1980,
%T 2016,2160,2184,2520,2772,3024,3420,3600,3780,4680,5040,5940,6048,
%U 6552,7440,7560,7800,8128,8190,8280,9240,9828,9900,10080,10530,11088,11400,13680
%N Freestyle perfect numbers n = Product_{i=1,..,k} f_i^e_i where 1 < f_1 < ... < f_k, e_i > 0, such that 2n = Product_{i=1,..,k} (f_i^(e_i+1)-1)/(f_i-1).
%C Only one odd freestyle perfect number is known: 198585576189, found by Descartes.
%C This sequence consists of perfect numbers A000396 and those which aren't, called spoof-perfect numbers A174292. Roughly said, a spoof-perfect number is a number that would be perfect if one or more of its composite factors were wrongly assumed to be prime, i.e., taken as a "spoof prime". - _Daniel Forgues_, Nov 15 2009 (slightly rephrased)
%C The right hand side of the second equation in the definition, 2n = ..., equals the sum of divisors sigma(n), if all of the f_i are distinct primes. If they aren't, there arise some ambiguities: See A174292 for further discussion. - _M. F. Hasler_, Jan 13 2013
%D R. K. Guy, Unsolved Problems in Number Theory, B1.
%H OEIS wiki, <a href="https://oeis.org/wiki/Freestyle_perfect_numbers">Freestyle perfect numbers</a>.
%e n = 60 = (3^1)*(4^1)*(5^1), s = 120 = [(3^2-1)*(4^2-1)*(5^2-1)]/[(3-1)*(4-1)*(5-1)]. s-n = 120-60 = n. So 60 is in the sequence.
%t r[s_, n_, f_] := Catch[If[n == 1, s == 1, Block[{p, e}, Do[e = 1; While[ Mod[n, p^e] == 0, r[s*(p^(e+1) - 1)/(p-1), n/p^e, p] && Throw@True; e++], {p, Select[Divisors@n, f < # &]}]]; False]];
%t spoofQ[n_] := r[1/2/n, n, 1] && DivisorSigma[-1, n] != 2;
%t perfectQ[n_] := DivisorSigma[1, n] == 2*n;
%t Select[Range[10^4], spoofQ[#] || perfectQ[#]&] (* _Jean-François Alcover_, May 16 2017, using _Giovanni Resta_'s code for A174292 *)
%Y Cf. A000396, A174292.
%K nonn,nice
%O 1,1
%A _Naohiro Nomoto_, Nov 13 2000
%E a(41)-a(45) from _Amiram Eldar_, Dec 27 2018
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