Revision #621 → #1467 · back to history
addedPerfect number03aa5a5cdd70
added6 is perfect03cfd0e747cf
added28 is perfectdc270f83c80d
addedFirst seven perfect numbersc041bd0fb551
addedAliquot sum0e5891532565
addedHalf-sum-of-divisors characterization1e651ad2d1a2
addedNicomachus's claim on perfect number form6bd4755f8622
addedCataldi: perfect numbers from Euclid's rule end in 6 or 8178a765eb87c
addedEuclid's even perfect number theorem7ccc80e98d67
addedFirst four perfect numbers from formulaebf999ca5afc
addedMersenne prime254a76496806
addedNecessary condition: exponent must be prime215f3dfcb0f0
added2047 is not prime84512627ec73
addedEuclid–Euler theorema252ea633869
addedFirst 50 even perfect numbers5dbdff68fcf2
addedEven perfect numbers are triangular and hexagonal5259f47415a5
addedCentered nonagonal and sum of odd cubes7dfeecf66352
addedTriangular form of even perfect numbers (except 6)d6179c692caa
addedDigital root of even perfect numbers is 1535ead203065
addedBinary representation of even perfect numbers095cf56177e6
addedEven perfect numbers are perniciousd9ad23e9833c
addedEven perfect numbers are practicalac4ee19bb820
addedPomerance heuristic against odd perfect numbersaf0486290aeb
addedPerfect numbers are harmonic divisor numbers91b3b31e1651
addedNecessary conditions on odd perfect numbers5849d76ddbf4
addedEuler's form of an odd perfect number50a39cbe01ce
addedNot all exponents ≡ 1 (mod 3)ece4604638cc
addedNot all exponents ≡ 2 (mod 5)74c4865fd4a4
addedSmallest prime factor bound under exponent congruences8e8f445ab4f0
addedGeneralized smallest-prime-factor bound951d0e3eb50f
addedBound for exponent pattern of ones and twos0b1426f9fdf3
addedExcluded exponent tuples83cc3e4c7828
addedExcluded equal exponents983a742531b4
addedDescartes's near-perfect numberd3b424c5da28
addedDescartes numbersb92d6d528923
addedMakowski: only n³+1 perfect number is 28ef0092b40ae0
addedGallardo: 28 is unique sum of two cubes36c83c1a8108
addedReciprocals of divisors sum to 245ea84c13370
addedNumber of divisors is even90c231b78517
addedPerfect numbers are Ore harmonic numbers5e6d151be953
addedEven perfect numbers are not trapezoidal50bd6fb9793b
addedThree types of non-trapezoidal numbers7f1b267aae39
addedDensity of perfect numbersaece8d67bba9
addedEven perfect numbers end in 6 or 287891fd1bd970
addedOnly square-free perfect number is 691d9e5a4f110
addedDeficient and abundant numbersde783fdd4fab
addedAmicable and sociable numbers8aa251431cca
addedPractical number37087049bab5
addedPerfect number as fixed point of restricted divisor function2a7bbb85d7a2
addedGranville numbersb0a4918ba28f
addedSemiperfect number4a229565fff0
addedWeird numbers1b220792b800