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%I #19 Apr 25 2020 13:17:19
%S 1,2,3,5,8,11,15,20,24,35,59,80,84,119,224,239,335,359,360,480,539,
%T 719,720,840,1079,1259,1260,1679,2519,4199,5039,5040,6720,7559,9360,
%U 10079,10080,15119,20159,25199,25200,27719,32759,43680,50399,55439,75599
%N 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 Those n where A092405(n) sets records.
%C 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 Amiram Eldar, <a href="/A175143/b175143.txt">Table of n, a(n) for n = 1..145</a>
%H 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 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 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 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 Cf. A000005, A002182, A067128, A092405.
%K nonn
%O 1,2
%A _Leroy Quet_, Feb 24 2010
%E Extended by _Ray Chandler_, Mar 05 2010
%E Terms beyond 80 from _R. J. Mathar_, Mar 05 2010
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