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A175143 a(1)=1. a(n) = the smallest integer > a(n-1) such that d(a(n))+d(a(n)+1) > d(a(n-1))+d(a(n-1)+1), where d(m) = the number of divisors of m.

%I A175143 #19 Apr 25 2020 13:17:19
%S A175143 1,2,3,5,8,11,15,20,24,35,59,80,84,119,224,239,335,359,360,480,539,
%T A175143 719,720,840,1079,1259,1260,1679,2519,4199,5039,5040,6720,7559,9360,
%U A175143 10079,10080,15119,20159,25199,25200,27719,32759,43680,50399,55439,75599
%N A175143 a(1)=1. a(n) = the smallest integer > a(n-1) such that d(a(n))+d(a(n)+1) > d(a(n-1))+d(a(n-1)+1), where d(m) = the number of divisors of m.
%C A175143 Those n where A092405(n) sets records.
%C A175143 Nicolas proved that: (1) Except for a finite number of terms, if k is in this sequence either k or k+1 is a largely composite number (A067128). (2) Except for a finite number of terms if k is a highly composite number (A002182) then k-1 is a term of this sequence. Apparently the only exceptions of (1) are 15, 80, 224, 6720, and 9360, and the only exceptions of (2) are 1, 24, 48, 180, 840, and 45360. - _Amiram Eldar_, Aug 24 2019
%H A175143 Amiram Eldar, <a href="/A175143/b175143.txt">Table of n, a(n) for n = 1..145</a>
%H A175143 Jean-Louis Nicolas, <a href="http://doi.org/10.4064/aa-49-4-395-412">Nombres hautement composés</a>, Acta Arithmetica, Vol. 49 (1988), pp. 395-412, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa49/aa4946.pdf">alternative link</a>. See p. 398.
%p A175143 A092405 := proc(n) numtheory[tau](n)+numtheory[tau](n+1) ; end proc: read("transforms") ; a092405 :=[seq(A092405(n),n=1..90000)] ; RECORDS(a092405)[2] ; # _R. J. Mathar_, Mar 05 2010
%t A175143 d1 = 1; dm = 0; s = {}; Do[d2 = DivisorSigma[0, n]; d = d1 + d2; If[d > dm, dm = d; AppendTo[s, n - 1]]; d1 = d2, {n, 2, 80000}]; s (* _Amiram Eldar_, Aug 24 2019 *)
%t A175143 smi[n_]:=Module[{k=n+1,ds=DivisorSigma[0,n]+DivisorSigma[0,n+1]},While[ DivisorSigma[ 0,k]+DivisorSigma[0,k+1]<=ds,k++];k]; NestList[smi,1,50] (* _Harvey P. Dale_, Apr 25 2020 *)
%Y A175143 Cf. A000005, A002182, A067128, A092405.
%K A175143 nonn
%O A175143 1,2
%A A175143 _Leroy Quet_, Feb 24 2010
%E A175143 Extended by _Ray Chandler_, Mar 05 2010
%E A175143 Terms beyond 80 from _R. J. Mathar_, Mar 05 2010