A135642 -id:A135642 - 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

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

A167853 Generalized mountain primes.

Original entry on oeis.org

2, 3, 5, 7, 131, 151, 163, 173, 181, 191, 193, 197, 241, 251, 263, 271, 281, 283, 293, 353, 373, 383, 397, 461, 463, 487, 491, 563, 571, 587, 593, 673, 683, 691, 787, 797, 1231, 1283, 1291, 1297, 1321, 1361, 1373, 1381, 1451, 1453, 1471

Offset: 1

Omar E. Pol, Nov 13 2009, Nov 15 2009

Primes in A134853. This sequence is finite because A134853 is.
Superset of A134951, mountain primes.
Question: How many terms are in this sequence?
The last term is a(7145) = 134567897654321. - Giovanni Resta, Mar 19 2013

			Illustration of 136973 as a generalized mountain prime:
. . . 9 . .
. . . . . .
. . . . 7 .
. . 6 . . .
. . . . . .
. . . . . .
. 3 . . . 3
. . . . . .
1 . . . . .

Giovanni Resta, Table of n, a(n) for n = 1..7145 (full sequence)

Cf. A000040, A134853, A134941, A134951, A135642, A167845.

More terms provided. Harvey P. Dale, Aug 19 2010

a(37)-a(47) corrected by Giovanni Resta, Mar 19 2013

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

A167845 Concave primes.

Original entry on oeis.org

131, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 233, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 331, 353, 367, 373, 379, 383, 389, 397, 431, 443, 461, 463, 467, 479, 487, 491, 499, 541, 563, 571, 577, 587, 593, 599, 641, 653, 661, 673

Offset: 1

Omar E. Pol, Nov 13 2009

Primes in A135642.

Primes whose 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.

Cf. A134811, A134951, A134971, A135642, A167841, A167842, A167843, A167844, A167846, A167853.

More terms from Max Alekseyev, Apr 24 2010

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

cp's OEIS Frontend

A135643 Straight-line numbers > 99.

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

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A135641 Convex numbers.

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A135600 Angled numbers with an internal digit as the vertex.

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A167853 Generalized mountain primes.

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A163278 Concave-convex numbers.

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