Search: id:A023199
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%I A023199
%S A023199 1,6,120,27720,122522400,130429015516800,1970992304700453905270400,
%T A023199 1897544233056092162003806758651798777216000,
%U A023199 4368924363354820808981210203132513655327781713900627249499856876120704000
%N A023199 a(n) = least k with sigma(k) >= nk.
%C A023199 Following a suggestion from Ed Pegg Jr, the sequence can be written in 
               a more readable form as: 1!, 3!, 5!, 11# * 3! * 2, 17# * 5! * 2, 
               29# * 7! * 4, 53# * 7! * 12, 89# * 11! * 2, 157# * 17# * 8! * 6, 
               271# * 23# * 10!, 487# * 29# * 10!, 857# * 37# * 11! * 42, 1487# 
               * 53# * 15! * 2, ..., where p# = primorial(p) = A034386.
%C A023199 Comment from T. D. Noe (noe(AT)sspectra.com), Jul 06 2005:
%C A023199 "Let c(p) be the smallest colossally-abundant number having the prime 
               factor p. See A073751 for info about computing these numbers.
%C A023199 Then the terms of this sequence can be expressed as
%C A023199 a(2) = c(3)
%C A023199 a(3) = c(5) * 2
%C A023199 a(4) = c(11) / 2
%C A023199 a(5) = c(17) / 3
%C A023199 a(6) = c(29) * 14
%C A023199 a(7) = c(53)
%C A023199 a(8) = c(89) * 4
%C A023199 a(9) = c(157) * 34
%C A023199 a(10) = c(271) * 23
%C A023199 a(11) = c(487) / 2
%C A023199 a(12) = c(857) / 2
%C A023199 a(13) = c(1487) * 212
%C A023199 a(14) = c(2621) * 710
%C A023199 a(15) = c(4567) * 2/21
%C A023199 a(16) = c(8011) / 2
%C A023199 a(17) = c(13999) * 1630"
%C A023199 Initially each term is divisible by the previous one. Is there a reason 
               why this should always be true? - Santi Spadaro (santi_spadaro(AT)virgilio.it), 
               Aug 13, 2002. The conjecture a(n)|a(n+1) holds out to n=10. - Devin 
               Kilminster (devin(AT)maths.uwa.edu.au), Mar 10 2003. The conjecture 
               a(n)|a(n+1) fails for n=15. - T. D. Noe (noe(AT)sspectra.com), Jul 
               08 2005.
%H A023199 Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">
               Abundancy : Some Resources </a>
%H A023199 Walter Nissen, <a href="http://upforthecount.com">Home Page</a> (listed 
               in lieu of email address)
%H A023199 T. D. Noe, <a href="http://www.sspectra.com/math/A023199.pdf">An algorithm 
               for finding the least k with sigma(k) >= nk</a>
%Y A023199 A subsequence of A004394. The dominating primes are in A108402.
%Y A023199 Sequence in context: A054479 A012475 A053777 this_sequence A007539 A040996 
               A110442
%Y A023199 Adjacent sequences: A023196 A023197 A023198 this_sequence A023200 A023201 
               A023202
%K A023199 nonn
%O A023199 1,2
%A A023199 David W. Wilson (davidwwilson(AT)comcast.net)
%E A023199 More terms from Walter Nissen Apr 15 1997. Further terms from Devin Kilminster 
               (devin(AT)maths.uwa.edu.au), Mar 10 2003
%E A023199 The term a(10) = 271#23#10! was apparently found independently by Bodo 
               Zinser and Don Reble, circa Jul 05 2005
%E A023199 The next term, a(11) = 487#29#10!, was corrected by Don Reble, Jul 06 
               2005
%E A023199 a(12) = 857#37#11!42 from Don Reble, Jul 06 2005
%E A023199 a(13) = 1487#53#15!2 found by T. D. Noe and confirmed by Don Reble, Jul 
               07 2005
%E A023199 a(14)-a(17) found by T. D. Noe and and rechecked by him Oct 11 2005
%E A023199 a(15) corrected. The conjecture still fails at n=15 T. D. Noe (noe(AT)sspectra.com), 
               Oct 13 2009

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