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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.
4
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
1,2
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
Jean-Louis Nicolas, Nombres hautement composés, Acta Arithmetica, Vol. 49 (1988), pp. 395-412, alternative link. See p. 398.
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Leroy Quet, Feb 24 2010
EXTENSIONS
Extended by Ray Chandler, Mar 05 2010
Terms beyond 80 from R. J. Mathar, Mar 05 2010
STATUS
approved