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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)
OFFSET

1,1

COMMENTS

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)

REFERENCES

R. K. Guy, Unsolved Problems in Number Theory, B1.

LINKS

Giovanni Resta, Table of n, a(n) for n = 1..500

OEIS wiki, Spoof perfect numbers.

C. Rivera (Ed.), Prime Puzzle 111. Spoof odd Perfect numbers, on primepuzzles.net (before 2000)

Giovanni Resta, Spoof factorizations of a(1)-a(500)

EXAMPLE

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.

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; Select[Range[10^4], spoofQ] (* Giovanni Resta, Feb 28 2013 *)

CROSSREFS

Cf. A058007, A000396; A222263, A222264.

Sequence in context: A178212 A182855 A009129 * A085987 A086974 A099831

Adjacent sequences:  A174289 A174290 A174291 * A174293 A174294 A174295

KEYWORD

nonn,nice

AUTHOR

Daniel Forgues, Nov 27 2010

EXTENSIONS

a(37)-a(44) from Amiram Eldar, Dec 27 2018

STATUS

approved

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Last modified April 17 10:51 EDT 2021. Contains 343064 sequences. (Running on oeis4.)