A001703 -id:A001703 - cp's OEIS frontend

A127422 Duplicate of A001703.

12, 123, 234, 345, 456, 567, 678, 789, 8910, 91011, 101112, 111213, 121314, 131415, 141516, 151617, 161718, 171819, 181920, 192021, 202122, 212223, 222324, 232425, 242526, 252627, 262728, 272829, 282930, 293031, 303132, 313233, 323334, 333435, 343536, 353637, 363738, 373839, 383940, 394041

Offset: 0

dead

A074991 Concatenation of n, n+1, n+2 divided by 3.

Original entry on oeis.org

4, 41, 78, 115, 152, 189, 226, 263, 2970, 30337, 33704, 37071, 40438, 43805, 47172, 50539, 53906, 57273, 60640, 64007, 67374, 70741, 74108, 77475, 80842, 84209, 87576, 90943, 94310, 97677, 101044, 104411, 107778, 111145, 114512

Offset: 0

Amarnath Murthy, Aug 31 2002

If all the three numbers have the same number of digits then the terms are in A.P. with a common difference that follows a pattern with increase in number of digits. E.g. for n = 1 to 7 the common difference, c.d. = 37. For n = 10 to 97 c.d. = 3367, For three digit numbers from 100 to 997 it is 333667 etc.

			a(4) = 456/3 = 152.

Cf. A074992.

Equals (1/3) A001703.

read("transforms") ;
A074991 := proc(n)
        digcatL([n,n+1,n+2]) ;
        %/3 ;
end proc:
seq(A074991(n),n=0..50) ; # R. J. Mathar, Oct 04 2011

f[n_] := FromDigits@ Flatten@ IntegerDigits[{n, n + 1, n + 2}]/3; Array[f, 26, 0] (* Robert G. Wilson v *)

Incorrect conjecture deleted by Colin Barker, Sep 26 2013

A035333 Concatenation of two or more consecutive positive integers.

Original entry on oeis.org

12, 23, 34, 45, 56, 67, 78, 89, 123, 234, 345, 456, 567, 678, 789, 910, 1011, 1112, 1213, 1234, 1314, 1415, 1516, 1617, 1718, 1819, 1920, 2021, 2122, 2223, 2324, 2345, 2425, 2526, 2627, 2728, 2829, 2930, 3031, 3132, 3233, 3334, 3435, 3456, 3536, 3637, 3738

Offset: 1

Erich Friedman

Paul Tek, Table of n, a(n) for n = 1..10000

For concatenations of exactly k consecutive integers see A000027 (k=1), A127421 (k=2), A001703 (k=3), A279204 (k=4).
See also A007908 for concatenation of 1 through n.
For primes see A052087.
All of A007908, A052087, A053067, A279610 are subsequences.

import heapq
from itertools import islice
def agen():
    c = 12
    h = [(c, 1, 2)]
    nextcount = 3
    while True:
        (v, s, l) = heapq.heappop(h)
        yield v
        if v >= c:
            c = int(str(c) + str(nextcount))
            heapq.heappush(h, (c, 1, nextcount))
            nextcount += 1
        l += 1; v = int(str(v)[len(str(s)):] + str(l)); s += 1
        heapq.heappush(h, (v, s, l))
print(list(islice(agen(), 47))) # Michael S. Branicky, Dec 23 2021

Edited by Charles R Greathouse IV, Apr 28 2010

Corrected by Paul Tek, Jun 08 2013

A127421 Numbers whose decimal expansion is a concatenation of 2 consecutive increasing nonnegative numbers.

Original entry on oeis.org

1, 12, 23, 34, 45, 56, 67, 78, 89, 910, 1011, 1112, 1213, 1314, 1415, 1516, 1617, 1718, 1819, 1920, 2021, 2122, 2223, 2324, 2425, 2526, 2627, 2728, 2829, 2930, 3031, 3132, 3233, 3334, 3435, 3536, 3637, 3738, 3839, 3940, 4041, 4142, 4243, 4344, 4445, 4546

Offset: 1

Artur Jasinski, Jan 14 2007

Primes in the sequence are in A030458. - Bruno Berselli, Mar 25 2015

a(n) always has an even number of digits unless n is in A103456. - Alonso del Arte, Oct 30 2019

			a(1) = "0,1" = 1.
a(13) = "12,13" = 1213.

N. J. A. Sloane, Table of n, a(n) for n = 1..25000

A variant of A001704.
For concatenations of exactly k consecutive integers see A000027 (k = 1), A127421 (k = 2), A001703 (k = 3), A279204 (k = 4). For 2 or more see A035333.
Cf. A030458, A127423, A127424, A127425, A103456.

[Seqint(Intseq(n+1) cat Intseq(n)): n in [0..50]]; // Bruno Berselli, Mar 25 2015

a:= n-> parse(cat(n-1, n)):
seq(a(n), n=1..55);  # Alois P. Heinz, Jul 05 2018

nMax = 49; digitsList = IntegerDigits[Range[0, nMax]]; Table[FromDigits[Flatten[{digitsList[[n]], digitsList[[n + 1]]}]], {n, nMax - 1}] (* Alonso del Arte, Oct 24 2019 *)
Table[FromDigits[Flatten[IntegerDigits/@{n,n+1}]],{n,0,50}] (* Harvey P. Dale, May 16 2020 *)

for n in range(100): print(int(str(n)+str(n+1))) # David F. Marrs, Sep 17 2018

val numerStrs = (0 to 49).map(Integer.toString(_)).toList
val concats = (numerStrs.dropRight(1)) zip (numerStrs.drop(1))
concats.map(x => Integer.parseInt(x.1 + x._2)) // _Alonso del Arte, Oct 24 2019

More terms from Joshua Zucker, May 15 2007

A279204 Numbers whose decimal expansion is a concatenation of 4 consecutive increasing nonnegative numbers.

Original entry on oeis.org

1234, 2345, 3456, 4567, 5678, 6789, 78910, 891011, 9101112, 10111213, 11121314, 12131415, 13141516, 14151617, 15161718, 16171819, 17181920, 18192021, 19202122, 20212223, 21222324, 22232425, 23242526, 24252627, 25262728, 26272829, 27282930, 28293031, 29303132, 30313233, 31323334

Offset: 1

N. J. A. Sloane, Dec 17 2016

Primes in this sequence are A030471. Are there infinitely many primes in the sequence? - Chai Wah Wu, Dec 17 2016

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A030471 (primes).

For concatenations of exactly k consecutive integers see A000027(k=1), A127421 (k=2), A001703 (k=3), A279204 (k=4). For 2 or more see A035333.

A279204[n_] := FromDigits[Flatten[IntegerDigits[Range[n, n + 3]]]];
Array[A279204, 50] (* Paolo Xausa, Aug 26 2024 *)

def A279204(n):
    return int(str(n)+str(n+1)+str(n+2)+str(n+3)) # Chai Wah Wu, Dec 17 2016

A030471 Primes which are concatenations of four consecutive numbers.

Original entry on oeis.org

4567, 14151617, 20212223, 34353637, 58596061, 64656667, 118119120121, 140141142143, 148149150151, 176177178179, 196197198199, 218219220221, 220221222223, 236237238239, 238239240241, 268269270271, 278279280281

Offset: 1

Patrick De Geest

There are no primes which are concatenations of three consecutive numbers (A001703), in fact all such concatenations are divisible by 3. - Zak Seidov, Oct 25 2014

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

Cf. A001703, A030470, A084551.

A279204[n_] := FromDigits[Flatten[IntegerDigits[Range[n, n + 3]]]];
Select[Array[A279204, 500], PrimeQ] (* Paolo Xausa, Aug 26 2024 *)

lista(nn) = for(n=1, nn, if (isprime(q=eval(concat(concat(concat(Str(n), Str(n+1)), Str(n+2)), Str(n+3)))), print1(q, ", "))); \\ Michel Marcus, Oct 26 2014

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    strs = ["1", "2", "3", "4"]
    for k in count(1):
        if isprime(t:=int("".join(strs))): yield t
        strs = strs[1:] + [str(k+4)]
print(list(islice(agen(), 20))) # Michael S. Branicky, Aug 26 2024

Edited by Charles R Greathouse IV, Apr 23 2010

A287747 Concatenation {n, n + 1,.., n + 9}.

Original entry on oeis.org

12345678910, 234567891011, 3456789101112, 45678910111213, 567891011121314, 6789101112131415, 78910111213141516, 891011121314151617, 9101112131415161718, 10111213141516171819, 11121314151617181920, 12131415161718192021, 13141516171819202122, 14151617181920212223

Offset: 1

XU Pingya, Jun 01 2017

For the position of primes in this sequence, see A287244.

Paolo Xausa, Table of n, a(n) for n = 1..1000

Cf. A001704, A001703, A279204, A287244.

A287747[n_] := FromDigits[Flatten[IntegerDigits[Range[n, n + 9]]]];
Array[A287747, 25] (* Paolo Xausa, Aug 26 2024 *)

A306643 Numbers that, for some x, are the concatenation of x, x+1 and x+2 and are divisible by at least two of x, x+1 and x+2.

Original entry on oeis.org

123, 234, 345, 456, 8910, 101112, 230231232

Offset: 1

J. M. Bergot and Robert Israel, Jun 03 2019

012 is not included because leading 0's are not allowed.
From Charlie Neder, Jun 05 2019: (Start)
If x = 10^k - 2, then x|x+1|x+2 (with | denoting concatenation) will be congruent to 22 modulo x, -9 modulo x+1, and 0 modulo x+2.
If x = 10^k - 1, then x|x+1|x+2 will be congruent to 12 modulo x and 1 modulo x+1.
Therefore, the only term such that x and x+2 have different lengths is 8910.
By reducing modulo x + {0,1,2} it can be shown that if at least two of x|10^k+2, x+1|10^2k-1, and x+2|2*10^2k-10^k are true - presuming x and x+2 are the same length - then x|x+1|x+2 is in this sequence. No further terms corresponding to x < 10^18. (End)
No further terms corresponding to x < 10^50. - Chai Wah Wu, Jun 19 2019

			230231232 is the concatenation of 230, 231 and 232 and is divisible by 231 and 232.

Subsequence of A001703.

Cf. A308527.

cat3:= proc(x)
  local t;
  t:= 10^length(x+2);
  x*(1 + t*(1+10^length(x+1)))+t+2
end proc:
f:= proc(x) local q,a,b;
  q:= cat3(x);
  a:= (q/x)::integer;
  b:= (q/(x+1))::integer;
  if a and b then return q elif not(a) and not(b) then return NULL fi;
  if (q/(x+2))::integer then q else NULL fi
end proc:
map(f, [$1..1000]);

for k in range(1,8):
  for x in range(10**(k-1),10**k-2): # will not find 8910
    if sum([not (10**k+2)%x,not (10**(2*k)-1)%(x+1),\
    not (2*10**(2*k)+10**k)%(x+2)]) >= 2:
      print(str(x)+str(x+1)+str(x+2)) # Charlie Neder, Jun 05 2019

A375481 Starting numbers for the terms in A035333.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 2, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 3, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 4, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 5, 57, 58

Offset: 1

Nicholas M. R. Frieler, Aug 17 2024

			a(19) = 12 since A035333(19) = 1213, the concatenation of 12 and 13.
a(20) =  1 since A035333(20) = 1234, the concatenation of 1, 2, 3 and 4.

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

For the terms generated by consecutive concatenations see A035333.For concatenations of various numbers of consecutive integers see A000027 (k=1), A127421 (k=2), A001703 (k=3), A279204 (k=4).For primes that are the concatenation of two or more consecutive integers see A052087.

ConsecutiveNumber[num_, numberOfNumbers_] := Module[{},
  If[numberOfNumbers == 1, Return[num]];
  FromDigits@(IntegerDigits[num]~Join~
     IntegerDigits@ConsecutiveNumber[num + 1, numberOfNumbers - 1])
 ]
ConsecutiveNumberDigits[maxDigits_] := Module[{numList = {}, c, d},
  Do[
   numList =
     numList~Join~(Association[# -> ConsecutiveNumber[#, c]] & /@
        Range[Power[10, d - 1], Power[10, d] - 1]);,
   {d, 1, Floor[maxDigits/2]}, {c, 2, Floor[maxDigits/d]}
   ];
  SortBy[Select[numList, Values[#][[1]] < Power[10, maxDigits] &],
   Values[#] &]
 ]
Keys[ConsecutiveNumberDigits[8]]//Flatten (* number in this line corresponds to the maximum number of digits of the concatenated terms *)

import heapq
from itertools import islice
def agen(): # generator of terms
    c = 12
    h = [(c, 1, 2)]
    nextcount = 3
    while True:
        (v, s, l) = heapq.heappop(h)
        yield s
        if v >= c:
            c = int(str(c) + str(nextcount))
            heapq.heappush(h, (c, 1, nextcount))
            nextcount += 1
        l += 1; v = int(str(v)[len(str(s)):] + str(l)); s += 1
        heapq.heappush(h, (v, s, l))
print(list(islice(agen(), 70))) # Michael S. Branicky, Aug 18 2024

A226041 Primes generated by concatenation of three consecutive numbers divided by three.

Original entry on oeis.org

41, 263, 50539, 64007, 151549, 191953, 205421, 232357, 252559, 266027, 272761, 306431, 326633, 35035369, 39039373, 52386053, 62396063, 70404071, 77077411, 82416083, 96430097, 103103437, 106440107, 110444111

Offset: 1

Jayanta Basu, May 24 2013

			41 is in the list since concatenation of 1, 2 and 3 gives 123/3 = 41. Similarly concatenation of 15, 16 and 17 gives 151617 and 151617/3 = 50539, a prime.

Cf. A001703.

Select[Table[FromDigits[Flatten[IntegerDigits[n + Range[-1, 1]]]]/3, {n, 350}], PrimeQ]

cp's OEIS Frontend

A127422 Duplicate of A001703.

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A074991 Concatenation of n, n+1, n+2 divided by 3.

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A035333 Concatenation of two or more consecutive positive integers.

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A127421 Numbers whose decimal expansion is a concatenation of 2 consecutive increasing nonnegative numbers.

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A279204 Numbers whose decimal expansion is a concatenation of 4 consecutive increasing nonnegative numbers.

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A030471 Primes which are concatenations of four consecutive numbers.

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A287747 Concatenation {n, n + 1,.., n + 9}.

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A306643 Numbers that, for some x, are the concatenation of x, x+1 and x+2 and are divisible by at least two of x, x+1 and x+2.

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