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%I #28 Mar 18 2022 00:02:30
%S 1,2,4,6,12,24,48,60,120,240,360,1680,2520,5040,10080,15120,25200,
%T 27720,55440,110880,166320,277200,720720,1441440,2162160,3603600,
%U 7207200,10810800,122522400,183783600,2327925600,3491888400,80313433200
%N Numbers k such that, for all m < k, d_i(k) <= d_i(m) for i=1 to Min(d(k),d(m)), where d_i(k) denotes the i-th smallest divisor of k.
%C The function d(k) (A000005) is the number of divisors of k.
%C The defining criterion for this sequence is a sufficient, but not necessary, condition for membership in A095849.
%C Subsequence of A002182. - _David Wasserman_, Jun 28 2007
%C Why is 720 not in the sequence? The divisors of 360 begin 1,2,3,4,5,6,8,9,10,12,15,18 (A018412) and the divisors of 720 begin 1,2,3,4,5,6,8,9,10,12,15,16 (A018609). - _J. Lowell_, Aug 23 2007 [Answer from _Don Reble_, Sep 11 2007: 720 is precluded by 420. (1,2,3,4,5,6,7,10,12,14,15,20,21,...) (A018444).]
%C Conjecture: If k is in this sequence, then so is the smallest number with k divisors. (This conjecture is definitely false for A002182 (k=840) and A019505 (k=240).) - _J. Lowell_, Jan 24 2008
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau Of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 844.
%H J. Britton, <a href="http://britton.disted.camosun.bc.ca/perfect/jbperfect.htm">Perfect Number Analyzer</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Table_of_divisors">Table of divisors</a>.
%e As k increases, the positive integer k=6 sets or ties the existing records for smallest first, second and third-smallest divisors (1, 2 and 3), as well as for its fourth-smallest (6). Since no smaller integer has more than three divisors, 6 is a term of this sequence.
%o (PARI) ge(va, vb) = {for(i=1, min(#va, #vb), if (va[i] > vb[i], return(0));); return(-1);}
%o isok(k) = {my(dk = divisors(k)); for (m=1, k-1, my(dm = divisors(m)); if (! ge(dk, dm), return(0));); return(1);} \\ _Michel Marcus_, Mar 16 2022
%Y Cf. A123258.
%K nonn
%O 1,2
%A _Matthew Vandermast_, Jun 10 2004
%E More terms from _David Wasserman_, Jun 28 2007
%E Definition corrected by _Ray Chandler_, May 05 2008