OFFSET
1,1
COMMENTS
Only one odd freestyle perfect number is known: 198585576189, found by Descartes.
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)
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
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, B1.
LINKS
Heath Davison Lunt, Table of n, a(n) for n = 1..478
OEIS wiki, Freestyle perfect numbers.
EXAMPLE
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.
From M. F. Hasler, May 28 2026: (Start)
a(4) = 84 = 2 * 6 * 7 = (2+1)*(6+1)*(7+1)/2: This is the first example where one f_i, f_1 = 2, divides another one, f_2 = 6.
a(5) = 90 = 2 * 9 * 5 = (2+1)*(9+1)*(5+1)/2.
a(6) = 120 = (2+1)*(4+1)*(3*5+1)/2 is another example similar to a(4) where the "assumed prime" f_2 = 4 is the square of f_1 = 2. Since this makes it somehow unrealistic to assume f_2 to be prime, one might consider a more restrictive definition of freestyle perfect numbers where f_i | f_j (or even gcd(f_i, f_j) > 1) would be excluded.
a(7) = 336 = 4^2 * 3 * 7 is in the sequence because (16+4+1)*(3+1)*(7+1) = 336*2. This is the first example with a composite factor to a higher power. (End)
MATHEMATICA
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]];
spoofQ[n_] := r[1/2/n, n, 1] && DivisorSigma[-1, n] != 2;
perfectQ[n_] := DivisorSigma[1, n] == 2*n;
Select[Range[10^4], spoofQ[#] || perfectQ[#]&] (* Jean-François Alcover, May 16 2017, using Giovanni Resta's code for A174292 *)
PROG
(PARI) select( {is_A058007(n, s=2*n, min_f=1)=sigma(n)==s || fordiv(n, d, d > min_f && !isprime(d) && for(e=1, oo, s%(d^(e+1)\(d-1)) || !is_A058007(n\d^e, s\(d^(e+1)\(d-1)), d) || return(1); n%d^(e+1) && break))}, [1..10^4]) \\ M. F. Hasler, May 28 2026
CROSSREFS
KEYWORD
nonn,nice,changed
AUTHOR
Naohiro Nomoto, Nov 13 2000
EXTENSIONS
a(41)-a(45) from Amiram Eldar, Dec 27 2018
STATUS
approved