A135641 Convex numbers.

100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 112, 113, 114, 115, 116, 117, 118, 119, 124, 125, 126, 127, 128, 129, 136, 137, 138, 139, 148, 149, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 211, 212, 213, 214, 215, 216, 217

Offset: 1

Author

Omar E. Pol, Nov 30 2007

Keywords

• nonn
• base

Comments

The structure of digits represents a convex function or a convex object. In the graphic representation the points are connected by imaginary line segments from left to right.

Examples

			The number 742235 is a convex number.
  . . . . . .
  . . . . . .
  7 . . . . .
  . . . . . .
  . . . . . 5
  . 4 . . . .
  . . . . 3 .
  . . 2 2 . .
  . . . . . .
  . . . . . .
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A135642 (concave numbers), A135643 (straight line numbers), A163278 (concave-convex numbers).

Programs

Mathematica

convexQ[n_] := With[{dd = IntegerDigits[n]}, AllTrue[SequencePosition[dd, {, , _}][[All, 1]], dd[[#]] + dd[[#+2]] > 2 dd[[#+1]]&]];
Select[Range[100, 300], convexQ] (* Jean-François Alcover, Nov 01 2018 *)

PARI

is(n) = my (d=digits(n), cvx=0, ccv=0, str=0); for (i=1, #d-2, my (x=d[i]+d[i+2]-2*d[i+1]); if (x>0, cvx++, x<0, ccv++, str++)); return (cvx>0 && ccv==0) \\ Rémy Sigrist, Aug 09 2017