A135643 -id:A135643 - cp's OEIS frontend

A135642 Concave numbers.

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

A135602 Right-angled numbers with an internal digit as the vertex.

Original entry on oeis.org

101, 121, 212, 232, 323, 343, 434, 454, 545, 565, 656, 676, 767, 787, 878, 898, 989, 1012, 1210, 1232, 2101, 2123, 2321, 2343, 3212, 3234, 3432, 3454, 4323, 4345, 4543, 4565, 5434, 5456, 5654, 5676, 6545, 6567, 6765, 6787, 7656, 7678, 7876, 7898, 8767, 8789, 8987

Offset: 1

Omar E. Pol, Dec 02 2007

The structure of digits represents a right angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right. This sequence is finite. The final term has 19 digits: 9876543210123456789.

			Illustration using the final term of this sequence:
  9 . . . . . . . . . . . . . . . . . 9
  . 8 . . . . . . . . . . . . . . . 8 .
  . . 7 . . . . . . . . . . . . . 7 . .
  . . . 6 . . . . . . . . . . . 6 . . .
  . . . . 5 . . . . . . . . . 5 . . . .
  . . . . . 4 . . . . . . . 4 . . . . .
  . . . . . . 3 . . . . . 3 . . . . . .
  . . . . . . . 2 . . . 2 . . . . . . .
  . . . . . . . . 1 . 1 . . . . . . . .
  . . . . . . . . . 0 . . . . . . . . .

Jens Kruse Andersen, Table of n, a(n) for n = 1..525

Cf. A134810, A134853, A134941, A134970, A135641, A135642, A135643, A135600, A135601, A135603.

ups = list(tuple(range(i, j)) for i in range(9) for j in range(i+2, 11))
s = set(L[::-1] + R[1:] for L in ups for R in ups if L[0] == R[0])
s |= set(L[:-1] + R[::-1] for L in ups for R in ups if L[-1] == R[-1])
afull = sorted(int("".join(map(str, t))) for t in s if t[0] != 0)
print(afull[:47]) # Michael S. Branicky, Aug 02 2022

A135601 Acute-angled numbers with an internal digit as the vertex.

Original entry on oeis.org

102, 103, 104, 105, 106, 107, 108, 109, 120, 130, 131, 132, 140, 141, 142, 143, 150, 151, 152, 153, 154, 160, 161, 162, 163, 164, 165, 170, 171, 172, 173, 174, 175, 176, 180, 181, 182, 183, 184, 185, 186, 187, 190, 191, 192, 193, 194, 195

Offset: 1

Omar E. Pol, Dec 02 2007

The structure of digits represents an acute angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right. This sequence is finite. The final term has 14 digits: 98765432102468.

			Illustration using the final term of this sequence:
  9 . . . . . . . . . . . . .
  . 8 . . . . . . . . . . . 8
  . . 7 . . . . . . . . . . .
  . . . 6 . . . . . . . . 6 .
  . . . . 5 . . . . . . . . .
  . . . . . 4 . . . . . 4 . .
  . . . . . . 3 . . . . . . .
  . . . . . . . 2 . . 2 . . .
  . . . . . . . . 1 . . . . .
  . . . . . . . . . 0 . . . .

Micah Manary, Table of n, a(n) for n = 1..2337
Micah Manary, Python program

Cf. A134810, A134853, A134941, A134970, A135641, A135642, A135643, A135600, A135602, A135603.

progressions = set(tuple(range(i, j+1, d)) for i in range(10) for d in range(1, 10-i) for j in range(i+d, 10))
s = set()
for left in progressions:
    for right in progressions:
        dl, dr = left[1] - left[0], right[1] - right[0]
        if dl + dr > 2:
            if left[-1] == right[-1]: s.add(left[:-1] + right[::-1])
            if left[0] == right[0]: s.add(left[::-1] + right[1:])
afull = sorted(int("".join(map(str, t))) for t in s if t[0] != 0)
print(afull[:53]) # Michael S. Branicky, Aug 02 2022

If a(n) does not end in 0, then A004086(a(n)) is a term; if a(n) does not start with 9, then A061601(a(n)) is a term. - Michael S. Branicky, Aug 02 2022

A135603 Obtuse-angled numbers with an internal digit as the vertex.

Original entry on oeis.org

100, 110, 112, 113, 114, 115, 116, 117, 118, 119, 122, 124, 125, 126, 127, 128, 129, 133, 134, 136, 137, 138, 139, 144, 145, 146, 148, 149, 155, 156, 157, 158, 166, 167, 168, 169, 177, 178, 179, 188, 189, 199, 200, 211, 220, 221, 223, 224, 225, 226, 227, 228, 229, 233

Offset: 1

Omar E. Pol, Dec 02 2007

The structure of digits represents an obtuse angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right.

For each k >= 11, there are 354 k-digit terms. - Michael S. Branicky, Aug 03 2022

			Illustration of the number 9753111:
  9 . . . . . .
  . . . . . . .
  . 7 . . . . .
  . . . . . . .
  . . 5 . . . .
  . . . . . . .
  . . . 3 . . .
  . . . . . . .
  . . . . 1 1 1
  . . . . . . .

Michael S. Branicky, Table of n, a(n) for n = 1..10232

Cf. A134810, A134853, A134941, A134970, A135641, A135642, A135643, A135600, A135601, A135602.

from itertools import count, islice
def ok3(n):
    if n < 100: return False
    d = list(map(int, str(n)))
    m1, m2 = (d[1]-d[0], d[-1]-d[-2])
    return len({m1, m2}) == 2 and m1*m2 >= 0
def agen():
    seeds = [k for k in range(100, 1000) if ok3(k)]
    for digits in count(4):
        yield from sorted(seeds)
        new, pow10 = set(), 10**(digits-1)
        for q in seeds:
            d = list(map(int, str(q)))
            e1, e2 = d[0] - (d[1]-d[0]), d[-1] + (d[-1]-d[-2])
            if 9 >= e1 > 0: new.add(e1*pow10 + q)
            if 9 >= e2 >= 0: new.add(10*q + e2)
        seeds = new
print(list(islice(agen(), 54))) # Michael S. Branicky, Aug 03 2022

a(49) and beyond from Michael S. Branicky, Aug 03 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

A135600 Angled numbers with an internal digit as the vertex.

Original entry on oeis.org

100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147

Offset: 1

Omar E. Pol, Dec 02 2007

The structure of digits represents an angle. The vertex is an internal digit. In the graphic representation the points are connected by imaginary line segments from left to right. The last acute-angled number of this sequence has 14 digits: 98765432102468. The last right-angled number of this sequence has 19 digits: 9876543210123456789. All 3-digit numbers are terms of this sequence. Next terms are 1000, 1012, 1024, 1036, 1048, 1110, 1111, 1112, 1113, 1114, ....

For each k >= 20, there are 363 k-digit terms: 354 obtuse-angled and 9 straight-angled.- Michael S. Branicky, Aug 03 2022

			The acute-angled number 12342 (see A135601):
  . . . . .
  . . . 4 .
  . . 3 . .
  . 2 . . 2
  1 . . . .
The right-angled number 12343 (see A135602):
  . . . . .
  . . . 4 .
  . . 3 . 3
  . 2 . . .
  1 . . . .
The obtuse-angled number 12344 (see A135603):
  . . . . .
  . . . 4 4
  . . 3 . .
  . 2 . . .
  1 . . . .
The straight-angled (or straight-line) number 12345 (see A135643):
  . . . . 5
  . . . 4 .
  . . 3 . .
  . 2 . . .
  1 . . . .

David A. Corneth, Table of n, a(n) for n = 1..10910
David A. Corneth, PARI program

Cf. A134810, A134853, A134941, A134970, A135641, A135642, A135643, A135601, A135602, A135603.

\\ See PARI link. David A. Corneth, Aug 02 2022

from itertools import count, islice
def agen():
    seeds = [k for k in range(100, 1000)]
    for digits in count(4):
        yield from sorted(seeds)
        new, pow10 = set(), 10**(digits-1)
        for q in seeds:
            d = list(map(int, str(q)))
            e1, e2 = d[0] - (d[1]-d[0]), d[-1] + (d[-1]-d[-2])
            if 9 >= e1 > 0: new.add(e1*pow10 + q)
            if 9 >= e2 >= 0: new.add(10*q + e2)
        seeds = new
print(list(islice(agen(), 50))) # Michael S. Branicky, Aug 03 2022

A138141 Numbers with digits in ascending order that differ exactly by 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 23, 34, 45, 56, 67, 78, 89, 123, 234, 345, 456, 567, 678, 789, 1234, 2345, 3456, 4567, 5678, 6789, 12345, 23456, 34567, 45678, 56789, 123456, 234567, 345678, 456789, 1234567, 2345678, 3456789, 12345678, 23456789, 123456789

Offset: 1

Omar E. Pol, Mar 19 2008

This finite sequence has 45 members. The last member is 123456789. There are 10-k members with k digits. See A052017 for primes in this sequence. All members with 3 or more digits are straight-line numbers A135643.

Cf. A052017, A059043, A135643, A138142.

Table[FromDigits/@Partition[Range[9],n,1],{n,9}]//Flatten (* Harvey P. Dale, Mar 19 2017 *)

a(n) = floor(((9*t^2 - 189*t + 18*n + 182) * (10^t - 1) - 18*t) / 162), where t = floor((21 - sqrt(369 - 8*n)) / 2). - Christopher J. Thomas, Feb 14 2024

A163278 Concave-convex numbers.

Original entry on oeis.org

1010, 1011, 1020, 1021, 1022, 1023, 1030, 1031, 1032, 1033, 1034, 1035, 1040, 1041, 1042, 1043, 1044, 1045, 1046, 1047, 1050, 1051, 1052, 1053, 1054, 1055, 1056, 1057, 1058, 1059, 1060, 1061, 1062, 1063, 1064, 1065, 1066, 1067, 1068, 1069, 1070, 1071

Offset: 1

Omar E. Pol, Oct 16 2009

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

			The number of this sequence (A163278) is a concave-convex number:
. . . . . .
. . . . . 8
. . . . 7 .
. 6 . . . .
. . . . . .
. . . . . .
. . 3 . . .
. . . 2 . .
1 . . . . .
. . . . . .

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

Cf. A135600, A135601, A135602, A135603, A135641, A135642, A135643.

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

A138142 Nonnegative numbers with digits in descending order that differ exactly by 1.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 21, 32, 43, 54, 65, 76, 87, 98, 210, 321, 432, 543, 654, 765, 876, 987, 3210, 4321, 5432, 6543, 7654, 8765, 9876, 43210, 54321, 65432, 76543, 87654, 98765, 543210, 654321, 765432, 876543, 987654

Offset: 1

Omar E. Pol, Mar 19 2008

This finite sequence has 55 members. The last member is 9876543210. There are 11-k members with k digits. See A052016 for primes in this sequence. All members with 3 or more digits are straight-line numbers A135643.

			Last 10 members of this finite sequence:
a(45)=6543210
a(46)=7654321
a(47)=8765432
a(48)=9876543
a(49)=76543210
a(50)=87654321
a(51)=98765432
a(52)=876543210
a(53)=987654321
a(54)=9876543210

Cf. A052016, A135643, A138141.

fQ[n_]:=Module[{id=IntegerDigits[n]}, n<10 || Union[Differences[id]]=={-1}]; Select[Range[0, 100000], fQ] (* Vladimir Joseph Stephan Orlovsky, Dec 29 2010 *)
Sort[Flatten[With[{t=Range[9,0,-1]},Table[FromDigits/@Partition[t,n,1],{n,10}]]]] (* Harvey P. Dale, Oct 31 2013 *)

Examples corrected by Omar E. Pol, Dec 06 2008

cp's OEIS Frontend

A135642 Concave numbers.

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A135602 Right-angled numbers with an internal digit as the vertex.

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A135601 Acute-angled numbers with an internal digit as the vertex.

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Formula

A135603 Obtuse-angled numbers with an internal digit as the vertex.

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Extensions

A135641 Convex numbers.

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A135600 Angled numbers with an internal digit as the vertex.

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A138141 Numbers with digits in ascending order that differ exactly by 1.

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A163278 Concave-convex numbers.

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