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%I M0573 #34 Jun 26 2017 19:19:52
%S 2,3,4,6,9,14,22,35,55,89,142,230,373,609,996,1637,2698,4461,7398,
%T 12301,20503,34253,57348,96198,161659,272124,458789,774616,1309627,
%U 2216968,3757384,6375166,10828012,18409028,31326514,53354259,90945529,155142139
%N Least number of distinct prime factors in even numbers having an abundancy index > n.
%C The abundancy index of a number k is sigma(k)/k. - _T. D. Noe_, May 08 2006
%C The first differences of this sequence, A005347, begin the same as the Fibonacci sequence A000045. - _T. D. Noe_, May 08 2006
%C a(n) = smallest k such that A002110(k)/A005867(k) > n. - _Artur Jasinski_, Nov 06 2008
%C For speed and accuracy, the second Mathematica program uses 30-digit real numbers and interval arithmetic.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H R. K. Guy, <a href="/A000081/a000081.pdf">Letter to N. J. A. Sloane, 1988-04-12</a> (annotated scanned copy)
%H R. K. Guy and N. J. A. Sloane, <a href="/A005180/a005180.pdf">Correspondence</a>, 1988.
%H Richard Laatsch, <a href="http://www.jstor.org/stable/2690424">Measuring the abundancy of integers</a>, Mathematics Magazine 59 (2) (1986) 84-92.
%t prod=1; k=0; Table[While[prod<=n, k++; prod=prod*Prime[k]/(Prime[k]-1)]; k, {n,2,25}] (* _T. D. Noe_, May 08 2006 *)
%t prod=Interval[1]; k=0; Table[While[Max[prod]<=n, k++; p=Prime[k]; prod=N[prod*p/(p-1),30]]; If[Min[prod]>n, k, "too few digits"], {n,2,38}]
%o (PARI) a(n)=my(s=1,k); forprime(p=2,, s*=p/(p-1); k++; if(s>n, return(k))) \\ _Charles R Greathouse IV_, Aug 20 2015
%Y Cf. A005580 (least number of distinct prime factors in odd numbers having an abundancy index > n).
%Y Cf. A023199 (least k such that sigma(k)/k >= n) - _T. D. Noe_, Sep 18 2008
%K nonn
%O 2,1
%A _N. J. A. Sloane_, _R. K. Guy_
%E Edited by _T. D. Noe_, May 08 2006
%E a(26) added by _T. D. Noe_, Sep 18 2008
%E Typo corrected by Vincent E. Yu (yu.vincent.e(AT)gmail.com), Aug 14 2009
%E a(27)-a(36) from Vincent E. Yu (yu.vincent.e(AT)gmail.com), Aug 14 2009
%E Comment corrected by _T. D. Noe_, Apr 04 2010
%E a(37)-a(39) from _T. D. Noe_, Nov 16 2010
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