A163278 -id:A163278 - cp's OEIS frontend

A135643 Straight-line numbers > 99.

111, 123, 135, 147, 159, 210, 222, 234, 246, 258, 321, 333, 345, 357, 369, 420, 432, 444, 456, 468, 531, 543, 555, 567, 579, 630, 642, 654, 666, 678, 741, 753, 765, 777, 789, 840, 852, 864, 876, 888, 951, 963, 975, 987, 999, 1111, 1234

Offset: 1

Omar E. Pol, Nov 30 2007, Dec 09 2008, Nov 14 2009

Numbers with more than two digits whose digits are in arithmetic progression. The structure of digits represents a straight line. In the graphic representation the points are connected by imaginary line segments. For a(1) to a(45) this sequence is equal to A034840. Each term of this sequence that is greater than 9876543210 is a repdigit number (A010785).

Note that the sequence of straight-line numbers starts: 10, 11, 12, ..., 98, 99, 111, 123, ... All 2-digit numbers are straight-line numbers, but here the numbers < 100 are omitted. - Omar E. Pol, Nov 14 2009

			The number 3579 is a straight-line number:
  . . . 9
  . . . .
  . . 7 .
  . . . .
  . 5 . .
  . . . .
  3 . . .
  . . . .
  . . . .
  . . . .

Jens Kruse Andersen, Table of n, a(n) for n = 1..168

Cf. A010785, A034840.
Cf. A135600, A135601, A135602, A135603, A135641, A135642, A163278, A167847. - Omar E. Pol, Nov 14 2009
Cf. A247616 (subsequence).

a135643 n = a135643_list !! (n-1)
a135643_list = filter f [100..] where
   f x = all (== 0) ws where
         ws = zipWith (-) (tail vs) vs
         vs = zipWith (-) (tail us) us
         us = map (read . return) $ show x
-- Reinhard Zumkeller, Sep 21 2014

Select[Range[100,1300],Length[Union[Differences[IntegerDigits[#]]]]==1&] (* Harvey P. Dale, May 09 2012 *)

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 && str>0) \\ Rémy Sigrist, Aug 09 2017

from itertools import count, islice
def agen():
    progressions = ["".join(map(str, range(i, j+1, d))) for i in range(10) for d in range(1, 10-i) for j in range(i+2*d, 10)]
    s =  [p for p in progressions if p[0] != "0"]          # up
    s += [p[::-1] for p in progressions]                   # down
    s += [d*i for d in "123456789" for i in range(3, 11)]  # flat
    yield from sorted(set(int(w) for w in s))
    yield from (int(f*d) for d in count(11) for f in "123456789")
print(list(islice(agen(), 178))) # Michael S. Branicky, Aug 03 2022

A135642 Concave numbers.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Nov 30 2007

The structure of digits represents a concave function or a concave object. In the graphic representation the points are connected by imaginary line segments from left to right.
Only strictly concave numbers are included in this sequence; the interpolation between at least one pair of digits must be strictly less than some intermediate digit. - Franklin T. Adams-Watters, Jan 26 2014
Also numbers where the second difference of consecutive digits is at most 0 and at least one of the second differences is negative. - David A. Corneth, Aug 02 2022

			The number 12221 is a concave number. Note that the number of this sequence (A135642) is also a concave number as shown below:
.
9     . . . . . .      . . . . . .
8     . . . . . .      . . . . . .
7     . . . . . .      . . . . . .
6     . . . x . .      . . . 6 . .
5     . . x . . .      . . 5 . . .
4     . . . . x .      . . . . 4 .
3     . x . . . .      . 3 . . . .
2     . . . . . x      . . . . . 2
1     x . . . . .      1 . . . . .
0     . . . . . .      . . . . . .
.
Another example is 1342. On the other hand, 3124 is not in the sequence, it's in A135641. 1234 is not in the sequence, it's in A135643. 1243 is not in the sequence, it's in A163278. - _Omar E. Pol_, Jan 29 2014

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

Cf. A135600, A135601, A135602, A135603, A135641, A135643, A163278.

concaveQ[n_] := With[{dd = IntegerDigits[n]}, AllTrue[SequencePosition[dd, {, , _}][[All, 1]], dd[[#]] + dd[[#+2]] < 2 dd[[#+1]]&]];
Select[Range[100, 200], concaveQ] (* Jean-François Alcover, Nov 01 2018 *)

{ isconcave(n) = my(t,r); t=eval(Vec(Str(n))); r=0; for(i=1, #t, for(j=i+2, #t, for(k=i+1, j-1, if( t[k]*(j-i) < t[i]*(j-k) + t[j]*(k-i), return(0)); if( t[k]*(j-i) > t[i]*(j-k) + t[j]*(k-i), r=1); ))); r } /* Franklin T. Adams-Watters and Max Alekseyev, Jan 30 2014 */

is(n) = if(n<100, return(0)); my(d=digits(n), v=vector(#d-2, i, d[i+2] - 2*d[i+1] + d[i])); v=Set(v); v[1] < 0 && v[#v] <= 0 \\ David A. Corneth, Aug 02 2022

A135641 Convex numbers.

Original entry on oeis.org

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

Omar E. Pol, Nov 30 2007

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.

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

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

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

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 *)

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

A167846 Concave-convex primes.

Original entry on oeis.org

1021, 1031, 1033, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, 1229, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423

Offset: 1

Omar E. Pol, Nov 13 2009

Primes in A163278.

Prime numbers with more than three digits that are not straight-line numbers (A135643), concave numbers (A135642) or convex numbers (A135641).

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

Cf. A134811, A134951, A134971, A135641, A135642, A135643, A163278, A167841, A167842, A167843, A167844, A167845, A167853.

More terms from Rémy Sigrist, May 22 2019

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A135643 Straight-line numbers > 99.

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A135642 Concave numbers.

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A135641 Convex numbers.

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A167846 Concave-convex primes.

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