The OEIS is supported by the many generous donors to the OEIS Foundation.



Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 60th year, we have over 367,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

Other ways to Give
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A174292 Spoof-perfect numbers: Freestyle perfect numbers (A058007) which are not perfect numbers (A000396). 5
60, 84, 90, 120, 336, 840, 924, 1008, 1080, 1260, 1320, 1440, 1680, 1980, 2016, 2160, 2184, 2520, 2772, 3024, 3420, 3600, 3780, 4680, 5040, 5940, 6048, 6552, 7440, 7560, 7800, 8190, 8280, 9240, 9828, 9900, 10080, 10530, 11088, 11400, 13680, 14040, 15120, 16380 (list; graph; refs; listen; history; text; internal format)
Spoof-perfect numbers are freestyle perfect numbers which are not perfect numbers.
Only one odd spoof perfect number is known: 198585576189, found by Descartes.
Assuming all integer factorizations were tried in the range [1..9900] in A058007, where I removed 6, 28, 496, 8128 from the list (I did not do the search for spoof perfect numbers myself, so the accuracy of my list depends on the accuracy of A058007's list.)
Roughly said, a spoof-perfect number is a number that would be perfect if some (one or more) of its composite factors were wrongly assumed to be prime, i.e., taken as a spoof prime.
Contribution from M. F. Hasler, Jan 13 2013: (Start)
I added "roughly said" to the above last phrase, since different interpretations of "would be perfect if some of its composite factors were wrongly assumed to be prime" are possible, and Descartes's example does not help to decide: (Notations are those from A058007, n = Sum (f_i)^(e_i).)
(a) If a spoof prime factor f_i is composite, may it have some of the smaller (spoof or true prime) f_j as factors or not? (In Descartes's example, this is not the case. And "assumed to be prime" could well imply that the answer is "no". But there is no such restriction in A058007.)
(b) If f_i is composite, is it required that e_i is the highest possible power, i.e., the (f_i)-valuation of n (or of n divided by all smaller f_j to the powers e_j)? (In Descartes's example this is the case. And if product(f_i^e_i) is to be a "prime factorization" of n, then it should be the case. But there is no such restriction in A058007. Note that this is not a consequence of (a), because the f_i could have common factors: e.g., even if f_1=21, f_2=35, f_3=45 are "wrongly assumed to be prime", then n=21*35*45 would have the (f_1)-valuation = 2, i.e., factorization n = f_1^2*75.)
(c) Is it reasonable to allow for even spoof primes f_i? (In Descartes's example this is not the case. And it seems somehow inconceivable that an even number be "wrongly assumed to be prime". But there is no such restriction in A058007.
Depending on the answer to each of these questions, "spoof-perfect numbers" as defined using "composite factors were wrongly assumed to be prime", could mean at least 8 different sequences. (End)
R. K. Guy, Unsolved Problems in Number Theory, B1.
C. Rivera (Ed.), Prime Puzzle 111. Spoof odd Perfect numbers, on primepuzzles.net (before 2000)
n = 60 = (3^1)*(4^1)*(5^1), s = 120 = (3^2-1)/(3-1) * (4^2-1)/(4-1) * (5^2-1)/(5-1): s-n = 120-60 = n, so 60 is in the sequence.
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; Select[Range[10^4], spoofQ] (* Giovanni Resta, Feb 28 2013 *)
Sequence in context: A182855 A350371 A009129 * A085987 A356413 A086974
Daniel Forgues, Nov 27 2010
a(37)-a(44) from Amiram Eldar, Dec 27 2018

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 4 07:54 EST 2023. Contains 367557 sequences. (Running on oeis4.)