cp's OEIS Frontend

A023199 a(n) is the least k with sigma(k) >= n*k.

%I A023199 #46 Aug 26 2025 04:54:20
%S A023199 1,6,120,27720,122522400,130429015516800,1970992304700453905270400,
%T A023199 1897544233056092162003806758651798777216000,
%U A023199 4368924363354820808981210203132513655327781713900627249499856876120704000
%N A023199 a(n) is the least k with sigma(k) >= n*k.
%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 From _T. D. Noe_, Jul 06 2005: (Start)
%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. (End)
%C A023199 Initially, each term is divisible by the previous one. Is there a reason this should always be true? - _Santi Spadaro_, Aug 13 2002
%C A023199 The conjecture a(n)|a(n+1) holds out to n=10. - Devin Kilminster (devin(AT)maths.uwa.edu.au), Mar 10 2003
%C A023199 The conjecture a(n)|a(n+1) fails for n=15. - _T. D. Noe_, Jul 08 2005
%C A023199 We have a(n) = min{A007539(n), A134716(n)}, and clearly A007539(n) != A134716(n) for every n. For what values of n is the former less than the latter? - _Jeppe Stig Nielsen_, Jun 16 2015
%H A023199 Amiram Eldar, <a href="/A023199/b023199.txt">Table of n, a(n) for n = 1..13</a>
%H A023199 Walter Nissen, <a href="http://upforthecount.com/math/abundance.html">Abundancy : Some Resources</a>, 2008-2010.
%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>, 2005-2009.
%o A023199 (PARI) a(n) = my(k=1); while (sigma(k)/k < n, k++); k; \\ _Michel Marcus_, Oct 07 2019
%Y A023199 A subsequence of A004394.
%Y A023199 The dominating primes are in A108402.
%Y A023199 Cf. A000203 (sigma), A007539, A034386, A073751, A134716.
%K A023199 nonn,changed
%O A023199 1,2
%A A023199 _David W. Wilson_
%E A023199 More terms from _Walter Nissen_, Apr 15 1997
%E A023199 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 rechecked by him Oct 11 2005
%E A023199 a(15) corrected. The conjecture still fails at n=15. - _T. D. Noe_, Oct 13 2009