A166323 Numbers in which all the digits are larger than the arithmetic mean of their two neighbors.

110, 120, 121, 122, 130, 131, 132, 133, 134, 140, 141, 142, 143, 144, 145, 146, 150, 151, 152, 153, 154, 155, 156, 157, 158, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185

Offset: 1

Author

Claudio Meller, Oct 11 2009

Keywords

• fini
• full
• nonn
• base
• easy

Comments

Last term of the sequence is a(3569) = 36899863.

The criterion on the arithmetic mean is only applied to the digits which have two neighbors, that is, not to the first and last digit.

If we look at the isolated digits d(i) of n = sum_i d(i)*10^(i-1) as function values defined over the regular abscissas of i=1,2,..., the criterion is equivalent to saying that the discrete approximation of the second derivative of the function is negative where defined, that is, the function is everywhere concave.

From Rick L. Shepherd, Feb 20 2013: (Start)

There are 405, 925, 1149, 781, 271, 38 terms of digit lengths 3 through 8, respectively, where 110, 1220, 13320, 134430, 1466530, 14677640 are the least and 998, 9986, 99863, 899863, 6899863, 36899863 are the greatest.

This sequence has the property that any three-digit-or-larger substring of a term's digit string is a term. Hence the fact that there is no nine-digit term proves there are no terms with even more digits. The sequence of terms that are not substrings of greater terms begins 110, 120, 121, 130, 131, 140, 141, 142, 150, 151, 152, 160, 161, 162, 163, 170, .... (End)

Links

• Nathaniel Johnston, Table of n, a(n) for n = 1..3569 (full sequence)

Crossrefs

Cf. A135642. [R. J. Mathar, Oct 14 2009]

Programs

Maple

isA166323 := proc(n) local d,k: d:=convert(n,base,10): for k from 2 to nops(d)-1 do if(2*d[k]<=d[k-1]+d[k+1])then return NULL: fi: od: return n: end: seq(isA166323(n),n=100..200); # Nathaniel Johnston, Jun 17 2011

Mathematica

dlamQ[n_]:=And@@(#[[2]]>(#[[1]]+#[[3]])/2&/@Partition[IntegerDigits[n],3,1])
Select[Range[100,200],dlamQ]  (* Harvey P. Dale, Feb 17 2011 *)

Extensions

keyword:base and most of the comment added by R. J. Mathar, Oct 14 2009