A167847 -id:A167847 - 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

A231588 Primes with decimal digits in arithmetic progression mod 10.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 109, 173, 197, 307, 383, 593, 727, 739, 937, 2963, 4567, 4703, 5791, 7159, 8147, 9371, 10987, 15937, 19753, 37159, 52963, 53197, 58147, 71593, 72727, 73951, 76543

Offset: 1

Paul Tek, Nov 11 2013

This sequence contains straight-line primes (A167847).

a(216) has 1012 digits. - Michael S. Branicky, Aug 05 2022

			(7,2,7,2,7,...) is an arithmetic progression mod 10, hence the prime number 72727 appears in this sequence.
(7,6,5,4,3,...) is an arithmetic progression mod 10, hence the prime number 76543 appears in this sequence.

Michael S. Branicky, Table of n, a(n) for n = 1..215 (terms 1..147 from Paul Tek)
Paul Tek, PARI program for this sequence

Cf. A167847.

Select[Prime[Range[PrimePi[76543]]], Length[Union[Mod[Differences[IntegerDigits[#]], 10]]] <= 1 &]

PARI
```
See Link section.
```

from sympy import isprime
from itertools import count, islice
def bgen():
    yield from [2, 3, 5, 7]
    yield from (int("".join(str((s0+i*r)%10) for i in range(d))) for d in count(2) for s0 in range(1, 10) for r in range(-s0, 10-s0))
def agen(): yield from filter(isprime, bgen())
print(list(islice(agen(), 52))) # Michael S. Branicky, Aug 05 2022

A302907 For any number m with decimal digits (d_1, ..., d_k), let s(m) be the area of the convex hull of the set of points { (i, d_i), i = 1..k }; a(n) = 2 * s(prime(n)) (where prime(n) denotes the n-th prime number).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 8, 10, 2, 4, 4, 2, 4, 2, 8, 2, 8, 4, 10, 4, 14, 16, 14, 10, 8, 1, 1, 5, 7, 1, 5, 5, 7, 1, 7, 1, 11, 5, 13, 11, 13, 10, 2, 4, 8, 2, 4, 2, 4, 4, 2, 2, 8, 2, 10, 4, 8, 5, 13, 11, 1

Offset: 1

Rémy Sigrist, Dec 16 2018

As in A167847 and in similar sequences, we map the digits of a number to a set of points and consider its graphical and geometrical properties.

			For n = 26:
- the 26th prime number is 101,
- the corresponding convex hull is as follows:
   (1,1) +-----+ (3,1)
          \   /
           \ /
            + (2,0)
- it has area 1, hence a(26) = 2.

Rémy Sigrist, Illustration of a(10000) (using Pick's theorem)
Rémy Sigrist, PARI program for A302907

Cf. A000040, A167847.

PARI
```
See Links section.
```

a(n) = 0 iff the n-th prime number belongs to A167847.

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A135643 Straight-line numbers > 99.

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A231588 Primes with decimal digits in arithmetic progression mod 10.

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Mathematica

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Python

A302907 For any number m with decimal digits (d_1, ..., d_k), let s(m) be the area of the convex hull of the set of points { (i, d_i), i = 1..k }; a(n) = 2 * s(prime(n)) (where prime(n) denotes the n-th prime number).

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