In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself.
The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but not weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2 + 4 + 6 = 12.
Infinitely many weird numbers exist.[3] For example, 70p is weird for all primesp ≥ 149. In fact, the set of weird numbers has positive asymptotic density.[4]
It is not known if any odd weird numbers exist. If so, they must be greater than 1021.[5]
Sidney Kravitz has shown that for k a positive integer, Q a prime exceeding 2k, and
also prime and greater than 2k, then
is a weird number.[6] With this formula, he found the large weird number
Primitive weird numbers
A property of weird numbers is that if n is weird, and p is a prime greater than the sum of divisors σ(n), then pn is also weird.[4] This leads to the definition of primitive weird numbers, i.e. weird numbers that are not a multiple of other weird numbers (sequence A002975 in the OEIS). There are only 24 primitive weird numbers smaller than a million, compared to 1765 weird numbers up to that limit. The construction of Kravitz yields primitive weird numbers, since all weird numbers of the form are primitive, but the existence of infinitely many k and Q which yield a prime R is not guaranteed. It is conjectured that there exist infinitely many primitive weird numbers, and Melfi has shown that the infiniteness of primitive weird numbers is a consequence of Cramér's conjecture.[7] Primitive weird numbers with as many as 16 prime factors and 14712 digits have been found.[8]
^ Benkoski, Stan (August–September 1972). "E2308 (in Problems and Solutions)". The American Mathematical Monthly. 79 (7): 774. doi:10.2307/2316276. JSTOR 2316276.
^Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. pp. 113–114. ISBN1-4020-4215-9. Zbl 1151.11300.
^ Kravitz, Sidney (1976). "A search for large weird numbers". Journal of Recreational Mathematics. Baywood Publishing. 9 (2): 82–85. Zbl 0365.10003.
^ Melfi, Giuseppe (2015). "On the conditional infiniteness of primitive weird numbers". Journal of Number Theory. Elsevier. 147: 508–514. doi:10.1016/j.jnt.2014.07.024.
^ Amato, Gianluca; Hasler, Maximilian; Melfi, Giuseppe; Parton, Maurizio (2019). "Primitive abundant and weird numbers with many prime factors". Journal of Number Theory. Elsevier. 201: 436–459. arXiv:1802.07178. doi:10.1016/j.jnt.2019.02.027. S2CID 119136924.
weird, number, term, weird, number, also, refers, phenomenon, complement, arithmetic, number, theory, weird, number, natural, number, that, abundant, semiperfect, euler, diagram, abundant, primitive, abundant, highly, abundant, superabundant, colossally, abund. The term weird number also refers to a phenomenon in two s complement arithmetic In number theory a weird number is a natural number that is abundant but not semiperfect 1 2 Euler diagram of abundant primitive abundant highly abundant superabundant colossally abundant highly composite superior highly composite weird and perfect numbers under 100 in relation to deficient and composite numbersIn other words the sum of the proper divisors divisors including 1 but not itself of the number is greater than the number but no subset of those divisors sums to the number itself Contents 1 Examples 2 Properties 2 1 Primitive weird numbers 3 See also 4 References 5 External linksExamples EditThe smallest weird number is 70 Its proper divisors are 1 2 5 7 10 14 and 35 these sum to 74 but no subset of these sums to 70 The number 12 for example is abundant but not weird because the proper divisors of 12 are 1 2 3 4 and 6 which sum to 16 but 2 4 6 12 The first few weird numbers are 70 836 4030 5830 7192 7912 9272 10430 10570 10792 10990 11410 11690 12110 12530 12670 13370 13510 13790 13930 14770 sequence A006037 in the OEIS Properties EditUnsolved problem in mathematics Are there any odd weird numbers more unsolved problems in mathematics Infinitely many weird numbers exist 3 For example 70p is weird for all primes p 149 In fact the set of weird numbers has positive asymptotic density 4 It is not known if any odd weird numbers exist If so they must be greater than 1021 5 Sidney Kravitz has shown that for k a positive integer Q a prime exceeding 2k and R 2 k Q Q 1 Q 1 2 k displaystyle R frac 2 k Q Q 1 Q 1 2 k also prime and greater than 2k then n 2 k 1 Q R displaystyle n 2 k 1 QR is a weird number 6 With this formula he found the large weird number n 2 56 2 61 1 153722867280912929 2 10 52 displaystyle n 2 56 cdot 2 61 1 cdot 153722867280912929 approx 2 cdot 10 52 Primitive weird numbers Edit A property of weird numbers is that if n is weird and p is a prime greater than the sum of divisors s n then pn is also weird 4 This leads to the definition of primitive weird numbers i e weird numbers that are not a multiple of other weird numbers sequence A002975 in the OEIS There are only 24 primitive weird numbers smaller than a million compared to 1765 weird numbers up to that limit The construction of Kravitz yields primitive weird numbers since all weird numbers of the form 2 k p q displaystyle 2 k pq are primitive but the existence of infinitely many k and Q which yield a prime R is not guaranteed It is conjectured that there exist infinitely many primitive weird numbers and Melfi has shown that the infiniteness of primitive weird numbers is a consequence of Cramer s conjecture 7 Primitive weird numbers with as many as 16 prime factors and 14712 digits have been found 8 See also EditUntouchable numberReferences Edit Benkoski Stan August September 1972 E2308 in Problems and Solutions The American Mathematical Monthly 79 7 774 doi 10 2307 2316276 JSTOR 2316276 Richard K Guy 2004 Unsolved Problems in Number Theory Springer Verlag ISBN 0 387 20860 7 OCLC 54611248 Section B2 Sandor Jozsef Mitrinovic Dragoslav S Crstici Borislav eds 2006 Handbook of number theory I Dordrecht Springer Verlag pp 113 114 ISBN 1 4020 4215 9 Zbl 1151 11300 a b Benkoski Stan Erdos Paul April 1974 On Weird and Pseudoperfect Numbers Mathematics of Computation 28 126 617 623 doi 10 2307 2005938 JSTOR 2005938 MR 0347726 Zbl 0279 10005 Sloane N J A ed Sequence A006037 Weird numbers abundant A005101 but not pseudoperfect A005835 The On Line Encyclopedia of Integer Sequences OEIS Foundation comments concerning odd weird numbers Kravitz Sidney 1976 A search for large weird numbers Journal of Recreational Mathematics Baywood Publishing 9 2 82 85 Zbl 0365 10003 Melfi Giuseppe 2015 On the conditional infiniteness of primitive weird numbers Journal of Number Theory Elsevier 147 508 514 doi 10 1016 j jnt 2014 07 024 Amato Gianluca Hasler Maximilian Melfi Giuseppe Parton Maurizio 2019 Primitive abundant and weird numbers with many prime factors Journal of Number Theory Elsevier 201 436 459 arXiv 1802 07178 doi 10 1016 j jnt 2019 02 027 S2CID 119136924 External links Edit Mathematics portalWeisstein Eric W Weird number MathWorld Retrieved from https en wikipedia org w index php title Weird number amp oldid 1122570276, wikipedia, wiki, book, books, library,
