A127421 -id:A127421 - cp's OEIS frontend

A001704 a(n) = n concatenated with n + 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

N. J. A. Sloane

See A030457 for the indices of prime terms in this sequence. - Reinhard Zumkeller, Jun 27 2015 [Simplified by Jianing Song, Jan 27 2019]

N. J. A. Sloane, Table of n, a(n) for n = 1..25000 (first 1000 terms from T. D. Noe)

See A127421 for a version with offset 0.

Cf. A215027, A248378, A030457.

a001704 n = a001704_list !! (n-1)
a001704_list = map read (zipWith (++) iss $ tail iss) :: [Integer]
               where iss = map show [1..]
-- Reinhard Zumkeller, Oct 07 2014

[Seqint(Intseq(n+1) cat Intseq(n)): n in [1..50]]; // Vincenzo Librandi, Jul 08 2018

f:=proc(i) i*10^(1+floor(evalf(log10(i+1), 10)))+i+1; end: # gives a(n) - N. J. A. Sloane, Aug 04 2012
# alternative Maple program:
a:= n-> parse(cat(n, n+1)):
seq(a(n), n=1..55);  # Alois P. Heinz, Jul 05 2018

Table[FromDigits@Flatten@IntegerDigits[{n, n + 1}], {n, 100}] (* T. D. Noe, Aug 09 2012 *)

a(n)=eval(Str(n,n+1)) \\ Charles R Greathouse IV, Jul 23 2016
(Emacs Lisp)
;; Concatenation
(defun A001704 (n) (string-to-int (concat (int-to-string n) (int-to-string (1+ n)))))
;; Formula
(defun A001704 (n) (1+ n (* n (expt 10 (1+ (floor (log (1+ n) 10)))))))
(mapcar '(lambda (n) (cons n (A001704 n))) '(1 2 3 10 11 12 99 999)) => ((1 . 12) (2 . 23) (3 . 34) (10 . 1011) (11 . 1112) (12 . 1213) (99 . 99100) (999 . 9991000))
; Tim Chambers, Jul 07 2018

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

val numerStrs = (1 to 50).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 and Jon E. Schoenfield, May 15 2007

A001703 Decimal concatenation of n, n+1, and n+2.

Original entry on oeis.org

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

Offset: 0

mag(AT)laurel.salles.entpe.fr

All terms are divisible by 3. Every third term starting with a(2) is divisible by 9. - Alonso del Arte, May 27 2013

			a(8) = 8910 since the three consecutive numbers starting with 8 are 8, 9, 10, and these concatenate to 8910. (This is the first term that differs from A193431).

T. D. Noe, Table of n, a(n) for n = 0..1000

Cf. A074991.
For concatenations of exactly k consecutive integers see A000027 (k=1), A127421 (k=2), A279204 (k=4). For 2 or more see A035333.
See also A127422, A127423, A127424.

read(transforms) :
A001703 := proc(n)
    digcatL([n,n+1,n+2]) ;
end proc:
seq(A001703(n),n=1..20) ; # R. J. Mathar, Mar 29 2017
# Third Maple program:
a:= n-> parse(cat(n, n+1, n+2)):
seq(a(n), n=0..50); # Alois P. Heinz, Mar 29 2017

concat3Nums[n_] := FromDigits@ Flatten@ IntegerDigits[{n, n + 1, n + 2}]; Array[concat3Nums, 25] (* Robert G. Wilson v *)

a(n)=eval(Str(n,n+1,n+2)) \\ Charles R Greathouse IV, Oct 08 2011

for n in range(100): print(int(str(n)+str(n+1)+str(n+2))) # David F. Marrs, Sep 18 2018

The portion of the sequence with all three numbers having d digits - i.e., n in 10^(d-1)..10^d-3 - is in arithmetic sequence: a(n) = (10^(2*d)+10^d+1)*n + (10^d+2). - Franklin T. Adams-Watters, Oct 07 2011

Initial term 12 added and offset changed to 0 at the suggestion of R. J. Mathar. - N. J. A. Sloane, Mar 29 2017

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

A127423 a(1) = 1; for n > 1, a(n) = n concatenated with n - 1.

Original entry on oeis.org

1, 21, 32, 43, 54, 65, 76, 87, 98, 109, 1110, 1211, 1312, 1413, 1514, 1615, 1716, 1817, 1918, 2019, 2120, 2221, 2322, 2423, 2524, 2625, 2726, 2827, 2928, 3029, 3130, 3231, 3332, 3433, 3534, 3635, 3736, 3837, 3938, 4039, 4140, 4241, 4342, 4443, 4544, 4645

Offset: 1

Artur Jasinski, Jan 14 2007

a(1) could also have been defined to be 10. In that case the initial terms would be 10, 21, 32, 43, 54, 65, 76, 87, 98, 109, 1110, 1211, 1312, 1413, 1514, 1615, 1716, 1817, 1918, 2019, 2120, 2221, 2322, 2423, 2524, 2625, 2726, 2827, 2928, 3029, 3130, 3231, 3332, 3433, ... (Comment added in case someone is searching for that sequence.) - N. J. A. Sloane, Aug 11 2018

A010051(a(A054211(n))) = 1. - Reinhard Zumkeller, Jun 27 2015

			a(12) = 1211 because 12 and 11 are two consecutive decreasing numbers.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A127421-A127424, A001704, A248378, A054211, A010051, A317947.

a127423 n = a127423_list !! (n-1)
a127423_list = 1 : map read (zipWith (++) (tail iss) iss) :: [Integer]
                   where iss = map show [1..]
-- Reinhard Zumkeller, Oct 07 2014

[Seqint(Intseq(n) cat Intseq(n+1)): n in [0..50]]; // Vincenzo Librandi, Nov 08 2016

c2:=proc(x,y) local s: s:=proc(m) nops(convert(m,base,10)) end: x*10^s(y)+y: end: seq(c2(n,n-1),n=1..53); # Emeric Deutsch, Mar 07 2007

Join[{1}, nxt[n_] := Module[{idn = IntegerDigits[n + 1], idn1 = IntegerDigits[n]}, FromDigits[Join[idn, idn1]]]; Array[nxt, 70]] (* Vincenzo Librandi, Nov 08 2016 *)
nxt[{n_, a_}] := {n + 1, (n + 1) * 10^IntegerLength[n] + n}; NestList[nxt, {1, 1}, 50][[All, 2]] (* Harvey P. Dale, Jan 04 2019 *)

a(n) = if (n==1, 1, eval(Str(n, n-1))); \\ Michel Marcus, Oct 14 2016

val numerStrs = (1 to 50).map(Integer.toString(_)).toList
val concats = (numerStrs.drop(1)) zip (numerStrs.dropRight(1))
concats.map(x => Integer.parseInt(x.1 + x._2)) // _Alonso del Arte, Oct 24 2019

More terms from Emeric Deutsch, Mar 07 2007

A074993 a(n) = floor((concatenation of n, n+1)/2).

Original entry on oeis.org

0, 6, 11, 17, 22, 28, 33, 39, 44, 455, 505, 556, 606, 657, 707, 758, 808, 859, 909, 960, 1010, 1061, 1111, 1162, 1212, 1263, 1313, 1364, 1414, 1465, 1515, 1566, 1616, 1667, 1717, 1768, 1818, 1869, 1919, 1970, 2020, 2071, 2121, 2172, 2222, 2273, 2323, 2374

Offset: 0

Amarnath Murthy, Aug 31 2002

The first differences follow a pattern. Odd-indexed terms and even-indexed terms form separate A.P.s with the same common difference for all n except n = 10^k -1. The corresponding common differences are the repunits = (10^(d+1)-1)/9 where d = the number of digits in n.

Cf. A001704, A074991, A074992, A074994, A074995, A074996, A074997, A073086, A074999, A127421.

cc[n_]:=Floor[FromDigits[Join[IntegerDigits[n],IntegerDigits[n+1]]]/2]; Array[cc,40,0] (* Harvey P. Dale, Nov 11 2011 *)

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

A127424 Numbers whose decimal expansion is a concatenation of 3 consecutive decreasing numbers.

Original entry on oeis.org

210, 321, 432, 543, 654, 765, 876, 987, 1098, 11109, 121110, 131211, 141312, 151413, 161514, 171615, 181716, 191817, 201918, 212019, 222120, 232221, 242322, 252423, 262524, 272625, 282726, 292827, 302928, 313029, 323130, 333231, 343332, 353433, 363534

Offset: 1

Artur Jasinski, Jan 14 2007

			a(13)=141312 because 14 and 13 and 12 are three consecutive decreasing numbers

Cf. A127421-A127424.

Table[FromDigits[Flatten[IntegerDigits[#]&/@Range[n,n-2,-1]]],{n,2,40}] (* Harvey P. Dale, Nov 28 2013 *)

Corrected and extended by Harvey P. Dale, Nov 28 2013

A317744 Prime numbers which result as a concatenation of a decimal number and its binary representation.

Original entry on oeis.org

11, 311, 5101, 131101, 2511001, 37100101, 51110011, 59111011, 731001001, 931011101, 971100001, 1191110111, 12910000001, 13110000011, 13710001001, 15310011001, 17310101101, 19311000001, 21311010101, 21511010111, 24711110111, 25511111111, 319100111111

Offset: 1

Philip Mizzi, Aug 05 2018

The decimal number plus its Hamming weight A000120 must not be divisible by 3. - M. F. Hasler, Apr 05 2024

			11 is in the sequence because the binary representation of 1 is 1 and the concatenation of 1 and 1 gives 11, which is prime.
931011101 is in the sequence because it is the concatenation of 93 and 1011101 (the binary representation of 93) and is prime.

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

Cf. A000040, A030458, A052087, A052088, A052089, A127421, A236551, A287300, A287310.

Select[Table[FromDigits[Join[IntegerDigits[n],IntegerDigits[n,2]]],{n,400}],PrimeQ] (* Harvey P. Dale, Jul 15 2020 *)

nb(n)=fromdigits(concat(n,binary(n)))
A317744_upto(N=666)=[p|n<-[1..N], is/*pseudo*/prime(p=nb(n))] \\ M. F. Hasler, Apr 05 2024

from sympy import isprime
def nbinn(n): return int(str(n)+bin(n)[2:])
def ok(n): return isprime(nbinn(n))
def aprefixupto(p): return [nbinn(k) for k in range(1, p+1, 2) if ok(k)]
print(aprefixupto(319)) # Michael S. Branicky, Dec 27 2020

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

cp's OEIS Frontend

A001704 a(n) = n concatenated with n + 1.

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A001703 Decimal concatenation of n, n+1, and n+2.

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A035333 Concatenation of two or more consecutive positive integers.

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A127423 a(1) = 1; for n > 1, a(n) = n concatenated with n - 1.

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A074993 a(n) = floor((concatenation of n, n+1)/2).

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

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A127424 Numbers whose decimal expansion is a concatenation of 3 consecutive decreasing numbers.

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