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.

1, 2, 3, 5, 8, 11, 15, 20, 24, 35, 59, 80, 84, 119, 224, 239, 335, 359, 360, 480, 539, 719, 720, 840, 1079, 1259, 1260, 1679, 2519, 4199, 5039, 5040, 6720, 7559, 9360, 10079, 10080, 15119, 20159, 25199, 25200, 27719, 32759, 43680, 50399, 55439, 75599

Offset: 1

Author

Leroy Quet, Feb 24 2010

Keywords

• nonn

Comments

Those n where A092405(n) sets records.

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..145
• Jean-Louis Nicolas, Nombres hautement composés, Acta Arithmetica, Vol. 49 (1988), pp. 395-412, alternative link. See p. 398.

Crossrefs

Cf. A000005, A002182, A067128, A092405.

Programs

Maple

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

Mathematica

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 *)
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 *)

Extensions

Extended by Ray Chandler, Mar 05 2010

Terms beyond 80 from R. J. Mathar, Mar 05 2010