A134810 -id:A134810 - cp's OEIS frontend

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

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

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

A134811 Giza primes.

Original entry on oeis.org

2, 3, 5, 7, 787, 34543, 345676543, 34567876543

Offset: 1

Omar E. Pol, Nov 30 2007

Giza numbers (A134810) that are prime numbers. For n > 4 the structure of digits represents a pyramid at Giza. Also the top of a mountain. This sequence has only these 8 terms in base 10. For more information, see A134810.

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

Chris K. Caldwell and G. L. Honaker, Jr; Prime Curios!, The Dictionary of Prime Number Trivia, CreateSpace (2009), p. 209.

G. L. Honaker, Jr. and Chris K. Caldwell, Prime Curios! 34567876543

Cf. A000040, A048398, A046479, A134810.

ups = Flatten[Table[Range[i, j - 1], {i, 1, 9}, {j, i + 1, 10}], 1];afull = Sort[  Map[ToExpression@StringJoin@Map[ToString, #[[;; -2]] ~Join~ Reverse[#]] &, ups]];Select[afull,PrimeQ] (* James C. McMahon, Apr 11 2025 *)

A000040 INTERSECT A134810. - Omar E. Pol, Mar 25 2011

Reference and link added by Omar E. Pol, Mar 25 2011

A173071 Palindromic mountain primes.

Original entry on oeis.org

131, 151, 181, 191, 12421, 12721, 12821, 13831, 13931, 14741, 17971, 1235321, 1245421, 1257521, 1268621, 1278721, 1456541, 1469641, 1489841, 1579751, 1589851, 123484321, 123494321, 123575321, 136797631, 167898761, 12345854321

Offset: 1

Omar E. Pol, Feb 09 2010

All terms have an odd number of digits. - Emeric Deutsch, Mar 09 2010

			a(6) = 12721; is a palindromic mountain prime.
. . . . .
. . . . .
. . 7 . .
. . . . .
. . . . .
. . . . .
. . . . .
. 2 . 2 .
1 . . . 1

Chai Wah Wu, Table of n, a(n) for n = 1..39

Cf. A002113, A002385, A134810, A134811, A134941, A134951.

a := proc (n) local rev, n1: rev := proc (n) local nn: nn := convert(n, base, 10): add(nn[j]*10^(nops(nn)-j), j = 1 .. nops(nn)) end proc: n1 := convert(n, base, 10): if n1[1]=1 and isprime(n) = true and rev(n) = n and n1[1] < n1[2] and n1[2] < n1[3] and n1[3] < n1[4] then n else end if end proc: seq(a(n), n = 1000000 .. 9999999); # this program works only for 7-digit numbers; easily adjustable for other (2k+1)-digit numbers # Emeric Deutsch, Mar 09 2010

from itertools import combinations
from gmpy2 import is_prime
A173071_list = []
for l in range(1,10):
    for i in combinations('23456789',l):
        s = '1'+''.join(i)
        p = int(s+s[l-1::-1])
        if is_prime(p):
            A173071_list.append(p) # Chai Wah Wu, Nov 05 2015

More terms from Emeric Deutsch, Mar 09 2010, corrected Mar 19 2010

a(22)-a(27) from Donovan Johnson, Jul 22 2010

A261618 Concatenation of n, n+1 and n.

Original entry on oeis.org

121, 232, 343, 454, 565, 676, 787, 898, 9109, 101110, 111211, 121312, 131413, 141514, 151615, 161716, 171817, 181918, 192019, 202120, 212221, 222322, 232423, 242524, 252625, 262726, 272827, 282928, 293029, 303130, 313231, 323332, 333433, 343534, 353635, 363736

Offset: 1

José de Jesús Camacho Medina, Sep 09 2015

787, 9109, 111211, 131413, ... are primes. - N. J. A. Sloane, Sep 21 2015

			a(1) = concatenation(1, 2, 1) = 121.
a(10) = concatenation(10, 11, 10) = 101110.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A134810, A068660.

[Seqint(Intseq(n) cat Intseq(n+1) cat Intseq(n)):n in [1..36]]; // Marius A. Burtea, Dec 29 2019

f:= n -> n + 10^(1+ilog10(n))*(n+1)+10^(2+ilog10(n)+ilog10(n+1))*n:
map(f, [$1..100]); # Robert Israel, Dec 29 2019

Map[FromDigits@ Join[#1, #2, #1] & @@ IntegerDigits[#] &, Partition[Range@ 37, 2, 1]] (* Michael De Vlieger, Dec 29 2019 *)

a(n) = eval(concat(Str(n), concat(Str(n+1), Str(n)))); \\ Michel Marcus, Sep 10 2015

More terms from Michel Marcus, Sep 10 2015

A134999 Triangle-shaped numbers.

Original entry on oeis.org

100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 148, 149, 150, 151

Offset: 1

Omar E. Pol, Dec 05 2007

The structure of digits represents a triangle. Three digits are the vertex. In the graphic representation the points are connected by imaginary line segments. All 3-digit numbers are terms of this sequence, except for the straight-line numbers A135643. For 4 or more digits the terms are 1000, 1011, 1012, 1024, 1034, 1036, ...

			The triangle number 1024:
  . . . 4
  . . . .
  . . 2 .
  1 . . .
  . 0 . .

Cf. A134810, A135600, A135601, A135602, A135603, A135643.

Not to be confused with the triangular numbers, A000217!

A173070 Palindromic mountain numbers.

Original entry on oeis.org

1, 121, 131, 141, 151, 161, 171, 181, 191, 12321, 12421, 12521, 12621, 12721, 12821, 12921, 13431, 13531, 13631, 13731, 13831, 13931, 14541, 14641, 14741, 14841, 14941, 15651, 15751, 15851, 15951, 16761, 16861, 16961, 17871, 17971, 18981

Offset: 1

Omar E. Pol, Feb 09 2010

There are 256 terms, the last of which is 12345678987654321. - Michael S. Branicky, Aug 04 2022

			13731 is in the sequence because it is a palindrome (A002113) and it is also a mountain number (A134941).
. . . . .
. . . . .
. . 7 . .
. . . . .
. . . . .
. . . . .
. 3 . 3 .
. . . . .
1 . . . 1

Michael S. Branicky, Table of n, a(n) for n = 1..256 (full sequence)

Cf. A002113, A134941, A134951, A134810, A134853, A173071

from itertools import chain, combinations as combs
def c(s): return s[0] == s[-1] == 1 and s == s[::-1]
ups = list(chain.from_iterable(combs(range(10), r) for r in range(2, 11)))
s = set(L[:-1] + R[::-1] for L in ups for R in ups if L[-1] == R[-1])
afull = [1] + sorted(int("".join(map(str, t))) for t in s if c(t))
print(afull[:40]) # Michael S. Branicky, Aug 04 2022

A193409 Crater numbers.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 101, 212, 323, 434, 545, 656, 767, 878, 989, 21012, 32123, 43234, 54345, 65456, 76567, 87678, 98789, 3210123, 4321234, 5432345, 6543456, 7654567, 8765678, 9876789, 432101234, 543212345, 654323456, 765434567, 876545678, 987656789, 54321012345, 65432123456, 76543234567, 87654345678, 98765456789, 6543210123456, 7654321234567, 8765432345678, 9876543456789, 765432101234567, 876543212345678, 987654323456789, 87654321012345678, 98765432123456789, 9876543210123456789

Offset: 1

Jaroslav Krizek, Jul 25 2011

For n>9 the structure of digits represents a crater. The first and last digit of each term are identical. The first digits are in consecutive decreasing order. The last digits are in consecutive increasing order. The numbers have only one smallest digit. The number of digits is odd. This sequence is finite with 55 terms. The final term is 9876543210123456789.
Finite subset of primes of this sequence: 2, 3, 5, 7, 101, 7654567.
There are 11 - k terms with 2*k - 1 digits. - Omar E. Pol, Aug 04 2011

			Illustration using a(32)=7654567:
  7  .  .  .  .  .  7
  .  6  .  .  .  6  .
  .  .  5  .  5  .  .
  .  .  .  4  .  .  .

Subset of palindromes (A002113), A193412 and valley numbers (A193413).

Cf. A134810, A134970. - Omar E. Pol, Aug 04 2011

Flatten[Table[FromDigits/@(Join[Reverse[Rest[#]],#]&/@Partition[ Range[ 0,9],n,1]),{n,10}]] (* Harvey P. Dale, Dec 27 2018 *)

ups = [tuple(range(i, j)) for i in range(10) for j in range(i+1, 11)]
afull = sorted(int("".join(map(str, u[::-1] + u[1:]))) for u in ups)
print(afull) # Michael S. Branicky, Aug 02 2022

Corrected and extended by Jaroslav Krizek, Jul 27 2011

cp's OEIS Frontend

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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Python

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A135600 Angled numbers with an internal digit as the vertex.

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PARI

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A134811 Giza primes.

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Mathematica

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A173071 Palindromic mountain primes.

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Maple

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A261618 Concatenation of n, n+1 and n.

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Magma

Maple