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A067128 Ramanujan's largely composite numbers, defined to be numbers m such that d(m) >= d(k) for k = 1 to m-1.

%I A067128 #79 Jun 27 2025 23:38:32
%S A067128 1,2,3,4,6,8,10,12,18,20,24,30,36,48,60,72,84,90,96,108,120,168,180,
%T A067128 240,336,360,420,480,504,540,600,630,660,672,720,840,1080,1260,1440,
%U A067128 1680,2160,2520,3360,3780,3960,4200,4320,4620,4680,5040,7560,9240
%N A067128 Ramanujan's largely composite numbers, defined to be numbers m such that d(m) >= d(k) for k = 1 to m-1.
%C A067128 This sequence is a subsequence of A034287; are they identical? They match for m up to 1500000.
%C A067128 Identical to A034287 for the 105834 terms less than 10^150.
%C A067128 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 A067128 By a result of Erdős (1944), a(n+1) <= 2*a(n): see Erdős link. - _David A. Corneth_, May 20 2016
%C A067128 It appears that if n > 13, then a(n) = A363658(n). - _Simon Jensen_, Aug 31 2023
%C A067128 Out of the first 10000 terms of this sequence, 1766 are adjacent to a prime. - _Dmitry Kamenetsky_, Jul 02 2024
%H A067128 T. D. Noe, <a href="/A067128/b067128.txt">Table of n, a(n) for n = 1..10000</a>
%H A067128 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 A067128 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 A067128 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 A067128 Vladimir Shevelev, <a href="http://arxiv.org/abs/1605.08884">On Erdős constant</a>, arXiv:1605.08884 [math.NT], 2016.
%e A067128 8 is a term as d(8) = 4 and d(k) <= 4 for k = 1,...,7.
%p A067128 isA067128 := proc(n)
%p A067128     local nd,k ;
%p A067128     nd := numtheory[tau](n) ;
%p A067128     for k from 1 to n-1 do
%p A067128         if numtheory[tau](k) > nd then
%p A067128             return false ;
%p A067128         end if;
%p A067128     end do:
%p A067128     true ;
%p A067128 end proc:
%p A067128 A067128 := proc(n)
%p A067128     option remember;
%p A067128     if n = 1 then
%p A067128         1;
%p A067128     else
%p A067128         for a from procname(n-1)+1 do
%p A067128             if isA067128(a) then
%p A067128                 return a;
%p A067128             end if;
%p A067128         end do:
%p A067128     end if;
%p A067128 end proc:
%p A067128 seq(A067128(n),n=1..60) ; # _R. J. Mathar_, Apr 15 2024
%t A067128 For[n=1; max=0, True, n++, If[(d=DivisorSigma[0, n])>=max, Print[n]; max=d]]
%t A067128 NestList[Function[last,
%t A067128   NestWhile[# + 1 &, last + 1,
%t A067128    DivisorSigma[0, #] < DivisorSigma[0, last] &]], 1, 70] (* _Steven Lu_, Nov 28 2022 *)
%o A067128 (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 A067128 For n with strictly increasing number of divisors, see A002182; A272314, A273011 (infinitary analog), subsequences A273015, A273016, A273018.
%Y A067128 Number of divisors of a(n): A273353.
%K A067128 easy,nonn
%O A067128 1,2
%A A067128 _Amarnath Murthy_, Jan 09 2002
%E A067128 Edited by _Dean Hickerson_, Jan 15 2002 and by _T. D. Noe_, Nov 07 2002