A135643 -id:A135643 - cp's OEIS frontend

A167847 Straight-line primes.

11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 4567, 76543, 23456789, 1111111111111111111, 11111111111111111111111

Offset: 1

Omar E. Pol, Nov 14 2009

Prime numbers with 2 digits together with the primes whose digits are in arithmetic progression. The structure of digits represents a straight line.
Note that in the graphic representation the points are connected by imaginary line segments (see also A135643).
Note that all two-digit primes are straight-line primes but this sequence has no three-digit terms.
No further terms between 23456789 and 115507867=prime(6600000). - R. J. Mathar, Dec 04 2009
All terms after 23456789 are repunit primes (A004022) with number of digits: 19, 23, 317, 1031, 49081, 86453, 109297, 270343, ... (A004023). - Jens Kruse Andersen, Jul 21 2014

			The number 4567 is straight-line prime:
  . . . .
  . . . .
  . . . 7
  . . 6 .
  . 5 . .
  4 . . .
  . . . .
  . . . .
  . . . .
  . . . .

Cf. A000040, A004022, A004023, A134811, A134951, A134971, A135643, A167841, A167842, A167843, A167844, A167845, A167846, A167853.

2 more terms from R. J. Mathar, Dec 04 2009

a(25)-a(26) from Jens Kruse Andersen, Jul 21 2014

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!

A247616 Numbers with more than two distinct digits in arithmetic progression.

Original entry on oeis.org

123, 135, 147, 159, 210, 234, 246, 258, 321, 345, 357, 369, 420, 432, 456, 468, 531, 543, 567, 579, 630, 642, 654, 678, 741, 753, 765, 789, 840, 852, 864, 876, 951, 963, 975, 987, 1234, 1357, 2345, 2468, 3210, 3456, 3579, 4321, 4567, 5432, 5678, 6420, 6543

Offset: 1

Reinhard Zumkeller, Sep 21 2014

A135643 without repdigit numbers (cf. A010785);

finite sequence with last and largest term a(96) = 9876543210.

			a(40) = 2468 with constant digit differences = +2;
a(41) = 3210 with constant digit differences = -1;
a(42) = 3456 with constant digit differences = +1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..96

Subsequence of A135643.

a247616 n = a247616_list !! (n-1)
a247616_list = filter f [100 .. 9876543210] where
   f x = head vs /= 0 && all (== 0) ws where
         ws = zipWith (-) (tail vs) vs
         vs = zipWith (-) (tail us) us
         us = map (read . return) $ show x

A301516 Numbers n with decimal expansion (d_1, ..., d_k) such that the convex hull of the set of points { (i, d_i), i = 1..k } has positive area.

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, 152, 153, 154, 155

Offset: 1

Rémy Sigrist, Dec 16 2018

This sequence is the complement of the straight-line numbers (0..99 alongside A135643).

This sequence first differs from A134999 at n = 857: a(857) = 1001 whereas A134999(857) = 1011.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A134999, A135643, A302907.

is(n, base=10) = my (d=digits(n, base)); for (i=1, #d-2, if (d[i]+d[i+2]-2*d[i+1], return (1))); return (0)

def ok(n):
    d = list(map(int, str(n)))
    return any(d[i]+d[i+2]-2*d[i+1] != 0 for i in range(len(d)-2))
print([k for k in range(1002) if ok(k)]) # Michael S. Branicky, Aug 03 2022 after Rémy Sigrist

A231701 Numbers > 100 with decimal digits in arithmetic progression mod 10.

Original entry on oeis.org

109, 111, 123, 135, 147, 159, 161, 173, 185, 197, 208, 210, 222, 234, 246, 258, 260, 272, 284, 296, 307, 319, 321, 333, 345, 357, 369, 371, 383, 395, 406, 418, 420, 432, 444, 456, 468, 470, 482, 494, 505, 517, 529, 531, 543, 555, 567, 579, 581, 593, 604, 616

Offset: 1

Paul Tek, Nov 12 2013

This sequence contains straight-line numbers > 99 (A135643).
Each set of numbers from 10^n to 10^(n+1) contains 90 of these numbers. - T. D. Noe, Nov 12 2013
The sequence mod 100 has period 900, the sequence mod 90 has period 8100. - Paul Tek, Nov 14 2013

			(8,2,6,0,4,...) is an arithmetic progression mod 10, hence the number 82604 appears in this sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Paul Tek)

Cf. A135643, A231588.

Select[Range[100, 10^3], Length[Union[Mod[Differences[IntegerDigits[#]], 10]]] <= 1 &] (* T. D. Noe, Nov 12 2013 *)

a(n) = my(len=3+(n-1)\90,   \
                 fs=10+((n-1)%90), \
                 f=fs\10,          \
                 s=fs%10);         \
               return(sum(i=1,len,10^(len-i)*((f+(i-1)*(s-f))%10)))

from itertools import count, islice
def agen(): yield from (int("".join(str((s0+i*r)%10) for i in range(d))) for d in count(3) for s0 in range(1, 10) for r in range(-s0, 10-s0))
print(list(islice(agen(), 52))) # Michael S. Branicky, Aug 05 2022

A135999 Numbers that are both triangular numbers A000217 and triangle-shaped numbers A134999.

Original entry on oeis.org

105, 120, 136, 153, 171, 190, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 703, 780, 820, 861, 903, 946, 990, 1596, 1711, 2145, 2346, 3240, 3321, 3486, 3570, 3741, 4371, 4560, 4753, 5565

Offset: 1

Omar E. Pol, Dec 10 2007

The number 496 appears to be the unique perfect number A000396 of this sequence. The interesting numbers 120 and 153 are supertriangular numbers.

			The numbers 120 as a triangle-shaped number A134999:
. 2 .
1 . .
. . 0
The number 120 as a triangular number A000217:
. . . . . . . O
. . . . . . .O O
. . . . . . O O O
. . . . . .O O O O
. . . . . O O O O O
. . . . .O O O O O O
. . . . O O O O O O O
. . . .O O O O O O O O
. . . O O O O O O O O O
. . .O O O O O O O O O O
. . O O O O O O O O O O O
. .O O O O O O O O O O O O
. O O O O O O O O O O O O O
.O O O O O O O O O O O O O O
O O O O O O O O O O O O O O O <--- A000217(15)=120.

Cf. A000217, A000396, A134999, A135600, A135601, A135602, A135603, A135643.

cp's OEIS Frontend

A167847 Straight-line primes.

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A134999 Triangle-shaped numbers.

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A247616 Numbers with more than two distinct digits in arithmetic progression.

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A301516 Numbers n with decimal expansion (d_1, ..., d_k) such that the convex hull of the set of points { (i, d_i), i = 1..k } has positive area.

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A231701 Numbers > 100 with decimal digits in arithmetic progression mod 10.

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Mathematica

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A135999 Numbers that are both triangular numbers A000217 and triangle-shaped numbers A134999.

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