Övernaturligt tal

Inom talteori är ett övernaturligt tal ett naturligt tal som är ymnigt men inte semiperfekt.[1][2] Med andra ord är ett övernaturligt tal ett tal vars summa av äkta delare är större än talet, men ingen delmängd av dessa delare har summan av talet.

De första övernaturliga talen är:

70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, 15610, 15890, 16030, 16310, 16730, 16870, 17272, 17570, 17990, 18410, 18830, 18970, 19390, 19670, … (talföljd A006037 i OEIS)

Egenskaper

Det har bevisats att det finns oändligt många övernaturliga tal;[3] följden av övernaturliga tal har till och med positiv asymptotisk densitet.[4]

Det är inte känt om det finns udda övernaturliga tal; om det finns ett sådant tal måste det vara större än 232 ≈ 4 × 109.[5]

Sidney Kravitz har bevisat att om k är ett positivt heltal, Q ett primtal större än 2k och om

;

också är ett primtal större än 2k, då är

ett övernaturligt tal.[6] Med denna formel upptäckte han det stora övernaturliga talet

.

Om n är övernaturligt och p är ett primtal större än sigmafunktionen σ(n) är pn också övernaturligt.[4]

Källor

Den här artikeln är helt eller delvis baserad på material från engelskspråkiga Wikipedia, Weird number, 19 april 2014.
  1. ^ Benkoski, Stan (Aug.-September 1972). ”E2308 (in Problems and Solutions)”. The American Mathematical Monthly 79 (7): sid. 774. doi:10.2307/2316276. 
  2. ^ Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. ISBN 0-387-20860-7. OCLC 54611248  Section B2.
  3. ^ Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, reds (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. sid. 113–114. ISBN 1-4020-4215-9 
  4. ^ [a b] Benkoski, Stan; Erdős, Paul (april 1974). ”On Weird and Pseudoperfect Numbers”. Mathematics of Computation 28 (126): sid. 617–623. doi:10.2307/2005938. 
  5. ^ Friedman, Charles N. (1993). ”Sums of divisors and Egyptian fractions”. J. Number Theory 44: sid. 328-339.  The result is attributed to "M. Mossinghoff at University of Texas - Austin".
  6. ^ Kravitz, Sidney (1976). ”A search for large weird numbers”. Journal of Recreational Mathematics (Baywood Publishing) 9 (2): sid. 82–85. 
v • r
Delbarhetsbaserade heltalsmängder
ÖversiktDelbarheten av 60
Faktoriserade former
Begränsade delarsummor
Med många delare
Alikvotföljdsrelaterade
Andra mängder
Lista över tal

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