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%I #66 Apr 15 2024 04:52:11
%S 1,2,3,4,6,8,10,12,18,20,24,30,36,48,60,72,84,90,96,108,120,168,180,
%T 240,336,360,420,480,504,540,600,630,660,672,720,840,1080,1260,1440,
%U 1680,2160,2520,3360,3780,3960,4200,4320,4620,4680,5040,7560,9240
%N Ramanujan's largely composite numbers, defined to be numbers m such that d(n) >= d(k) for k = 1 to m-1.
%C This sequence is a subsequence of A034287; are they identical? They match for m up to 1500000.
%C Identical to A034287 for the 105834 terms less than 10^150.
%C Every subsequence of terms, having the fixed greatest prime divisor prime(k), k=1,2,..., is finite. For a proof see A273015. The list of these subsequences begins {2,4,8}, {3,6,12,18,24,36,48,72,96,108}, ... - _Vladimir Shevelev_, May 13 2016
%C By a result of Erdős (1944), a(n+1) <= 2*a(n): see Erdős link. - _David A. Corneth_, May 20, 2016
%C It appears that if n > 13, then a(n) = A363658(n). - _Simon Jensen_, Aug 31 2023
%H T. D. Noe, <a href="/A067128/b067128.txt">Table of n, a(n) for n = 1..10000</a>
%H P. Erdős, <a href="https://www.renyi.hu/~p_erdos/1944-04.pdf">On Highly composite numbers</a>, J. London Math. Soc. 19 (1944), 130--133 MR7,145d; Zentralblatt 61,79.
%H Jean-Louis Nicolas, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa34/aa34410.pdf">Répartition des nombres largement composés,</a> Acta Arithmetica 34 (1979), 379-390.
%H J.-L. Nicolas and G. Robin, <a href="http://dx.doi.org/10.1023/A:1009764017495">Highly Composite Numbers by Srinivasa Ramanujan</a>, The Ramanujan Journal, Vol. 1(2), pp. 119-153, Kluwer Academics Pub.
%H Vladimir Shevelev, <a href="http://arxiv.org/abs/1605.08884">On Erdős constant</a>, arXiv:1605.08884 [math.NT], 2016.
%e 8 is a term as d(8) = 4 and d(k) <= 4 for k = 1,...,7.
%p isA067128 := proc(n)
%p local nd,k ;
%p nd := numtheory[tau](n) ;
%p for k from 1 to n-1 do
%p if numtheory[tau](k) > nd then
%p return false ;
%p end if;
%p end do:
%p true ;
%p end proc:
%p A067128 := proc(n)
%p option remember;
%p if n = 1 then
%p 1;
%p else
%p for a from procname(n-1)+1 do
%p if isA067128(a) then
%p return a;
%p end if;
%p end do:
%p end if;
%p end proc:
%p seq(A067128(n),n=1..60) ; # _R. J. Mathar_, Apr 15 2024
%t For[n=1; max=0, True, n++, If[(d=DivisorSigma[0, n])>=max, Print[n]; max=d]]
%t NestList[Function[last,
%t NestWhile[# + 1 &, last + 1,
%t DivisorSigma[0, #] < DivisorSigma[0, last] &]], 1, 70] (* _Steven Lu_, Nov 28 2022 *)
%o (PARI) is(n) = my(nd=numdiv(n)); for(k=1, n-1, if(numdiv(k) > nd, return(0))); return(1) \\ _Felix Fröhlich_, May 22 2016
%Y For n with strictly increasing number of divisors, see A002182; A272314, A273011 (infinitary analog), subsequences A273015, A273016, A273018.
%Y Number of divisors of a(n): A273353.
%K easy,nonn,changed
%O 1,2
%A _Amarnath Murthy_, Jan 09 2002
%E Edited by _Dean Hickerson_, Jan 15 2002 and by _T. D. Noe_, Nov 07 2002
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