A001704 -id:A001704 - cp's OEIS frontend

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

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

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 *)

A030457 Numbers k such that k concatenated with k+1 is prime.

Original entry on oeis.org

2, 6, 8, 12, 36, 42, 50, 56, 62, 68, 78, 80, 90, 92, 96, 102, 108, 120, 126, 138, 150, 156, 180, 186, 188, 192, 200, 216, 242, 246, 252, 270, 276, 278, 300, 308, 312, 318, 330, 338, 342, 350, 362, 368, 378, 390, 402, 410, 416, 420, 426, 428, 432

Offset: 1

Patrick De Geest

k is not congruent to 1 (mod 2), 1 (mod 3), or 4 (mod 5). - Charles R Greathouse IV, Apr 16 2012

			1213 is prime, therefore 12 is a term.

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

Cf. A030458, A054211, A052089, A052087, A052088.
Cf. A010051, A001704, A068700 (subsequence).
Numbers k such that k concatenated with k+m is prime: this sequence (m=1), A032617 (m=2), A032618 (m=3), A032619 (m=4), A032620 (m=5), A032621 (m=6), A032622 (m=7), A032623 (m=8), A032624 (m=9).

a030457 n = a030457_list !! (n-1)
a030457_list = filter ((== 1) . a010051' . a001704) [1..]
-- Reinhard Zumkeller, Jun 27 2015, Apr 26 2011

[n: n in [1..500] | IsPrime(Seqint(Intseq(n+1) cat Intseq(n)))]; // Vincenzo Librandi, Jul 23 2016

concat:=proc(a,b) local bb: bb:=nops(convert(b,base,10)): 10^bb*a+b end proc: a:=proc(n) if isprime(concat(n,n+1))=true then n else end if end proc: seq(a(n),n=0..500); # Emeric Deutsch, Nov 23 2007

Select[ Range[500], PrimeQ[ ToExpression[ StringJoin[ ToString[#], ToString[#+1]]]]&] (* Jean-François Alcover, Nov 18 2011 *)
Select[Range[500],PrimeQ[FromDigits[Join[IntegerDigits[#], IntegerDigits[ #+1]]]]&] (* Harvey P. Dale, Dec 23 2015 *)
Position[#[[1]]*10^IntegerLength[#[[2]]]+#[[2]]&/@Partition[Range[ 500], 2,1],?PrimeQ]//Flatten (* _Harvey P. Dale, Jul 14 2019 *)

for(n=1,10^5,if(isprime(eval(concat(Str(n),n+1))),print1(n,", "))); /* Joerg Arndt, Apr 27 2011 */

from sympy import isprime
def ok(n): return isprime(int(str(n)+str(n+1)))
print([k for k in range(500) if ok(k)]) # Michael S. Branicky, Apr 19 2023

A136414 Put the natural numbers together without spaces and read them two at a time advancing one space each time.

Original entry on oeis.org

12, 23, 34, 45, 56, 67, 78, 89, 91, 10, 1, 11, 11, 12, 21, 13, 31, 14, 41, 15, 51, 16, 61, 17, 71, 18, 81, 19, 92, 20, 2, 21, 12, 22, 22, 23, 32, 24, 42, 25, 52, 26, 62, 27, 72, 28, 82, 29, 93, 30, 3, 31, 13, 32, 23, 33, 33, 34, 43, 35, 53, 36, 63, 37, 73, 38, 83, 39, 94, 40, 4, 41, 14

Offset: 1

Ben Paul Thurston, Mar 31 2008

a(n) = A162711(n,2) for n>1. - Reinhard Zumkeller, Jul 11 2009

			34 is the third entry because the natural numbers written together look like 1234567891011 and reading them off two at a time produces 12, 23, 34, ...

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

Cf. A001704, A193431, A193492, A193493.

a136414 n = a136414_list !! (n-1)
a136414_list = zipWith (+) (tail a007376_list) $ map (10 *) a007376_list
-- Reinhard Zumkeller, Jul 28 2011

a(n) = 10*A007376(n) + A007376(n+1). - Reinhard Zumkeller, Jul 11 2009

More terms from Reinhard Zumkeller, Jul 11 2009

A248378 a(2n) = concatenation of n+1 with n+2, a(2n+1) = concatenation of n+2 with n+1.

Original entry on oeis.org

12, 21, 23, 32, 34, 43, 45, 54, 56, 65, 67, 76, 78, 87, 89, 98, 910, 109, 1011, 1110, 1112, 1211, 1213, 1312, 1314, 1413, 1415, 1514, 1516, 1615, 1617, 1716, 1718, 1817, 1819, 1918, 1920, 2019, 2021, 2120, 2122, 2221, 2223, 2322, 2324, 2423, 2425, 2524, 2526

Offset: 0

N. J. A. Sloane, Oct 07 2014

Charles Duvall, Email to N. J. A. Sloane, Oct 07 2014.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A001704, A127423.

Cf. A154530 (primes).

import Data.List (transpose)
a248378 n = a248378_list !! n
a248378_list = concat $ transpose [a001704_list, tail a127423_list]
-- Reinhard Zumkeller, Oct 07 2014

Table[{FromDigits[Join[IntegerDigits[n],IntegerDigits[n+1]]],FromDigits[Join[IntegerDigits[n+1],IntegerDigits[ n]]]},{n,30}]//Flatten (* Harvey P. Dale, Apr 19 2024 *)

A116283 k times k+7 gives the concatenation of two numbers m and m-1.

Original entry on oeis.org

7, 30, 64, 42753, 57241, 75423, 425072, 574922, 979528, 4301393, 5698601, 7028666, 4925000747, 5074999247, 7748266574, 8511881484, 8814851184, 7059602159672, 7106167933828, 7439286611621, 7485852385777

Offset: 1

Giovanni Resta, Feb 06 2006

Cf. A001704, A028563.

Cf. A116152, A116269, A116282, A116284, A116291.

def ok(n):
    s = str(n*(n+7)); h = (len(s)+1)//2; return int(s[:h])-1 == int(s[h:])
print(list(filter(ok, range(2, 10**6)))) # Michael S. Branicky, Jul 30 2021

A215027 a(n+1) = (concatenation of n and n+1) - a(n), a(0) = 0.

Original entry on oeis.org

0, 1, 11, 12, 22, 23, 33, 34, 44, 45, 865, 146, 966, 247, 1067, 348, 1168, 449, 1269, 550, 1370, 651, 1471, 752, 1572, 853, 1673, 954, 1774, 1055, 1875, 1156, 1976, 1257, 2077, 1358, 2178, 1459, 2279, 1560, 2380, 1661, 2481, 1762, 2582, 1863, 2683, 1964, 2784, 2065, 2885, 2166, 2986, 2267, 3087, 2368, 3188, 2469, 3289, 2570, 3390, 2671, 3491, 2772, 3592, 2873, 3693

Offset: 0

N. J. A. Sloane, Aug 04 2012, based on a posting to the Sequence Fans Mailing List by Eric Angelini

Eric Angelini defined this by saying that "a(n)+a(n+1) = concatenation of n and (n+1)".

An easy induction argument shows that a(n) is always positive.

			a(100) = concat(99,100) - a(99) = 99 100 - 4590 = 94510.

Cf. A001704, A215028.

f:=proc(i) i*10^(1+floor(evalf(log10(i+1), 10)))+i+1; end: # A001704
a:=proc(n) option remember; global f; if n=1 then 1 else f(n-1)-a(n-1); fi; end;

A215027(n,print_all=0)={my(a=print_all & print1(0));for(n=1,n,a=(n-1)*10^#Str(n)+n-a;print_all & print1(","a));a} \\ - M. F. Hasler, Aug 23 2012

The o.g.f. x*(1+10*x+810*x^9-720*x^10)/(1+x)/(1-x)^2 yields correct terms up to a(99), but not beyond. - M. F. Hasler, Aug 23 2012

Initial term a(0)=0 added by M. F. Hasler, Aug 23 2012

A032610 Concatenation of n and n + 5 or {n,n+5}.

Original entry on oeis.org

16, 27, 38, 49, 510, 611, 712, 813, 914, 1015, 1116, 1217, 1318, 1419, 1520, 1621, 1722, 1823, 1924, 2025, 2126, 2227, 2328, 2429, 2530, 2631, 2732, 2833, 2934, 3035, 3136, 3237, 3338, 3439, 3540, 3641, 3742, 3843, 3944, 4045, 4146, 4247

Offset: 1

Patrick De Geest, May 15 1998

Cf. A001704, A020338.

A032612 Concatenation of n and n+7.

Original entry on oeis.org

18, 29, 310, 411, 512, 613, 714, 815, 916, 1017, 1118, 1219, 1320, 1421, 1522, 1623, 1724, 1825, 1926, 2027, 2128, 2229, 2330, 2431, 2532, 2633, 2734, 2835, 2936, 3037, 3138, 3239, 3340, 3441, 3542, 3643, 3744, 3845, 3946, 4047, 4148, 4249

Offset: 1

Patrick De Geest, May 15 1998

Cf. A001704, A020338.

Table[n*10^IntegerLength[n+7]+n+7,{n,50}] (* Harvey P. Dale, Apr 26 2025 *)

def A032612(n): return n*(10**len(str(n+7))+1)+7 # Chai Wah Wu, Jun 29 2025

cp's OEIS Frontend

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

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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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A030457 Numbers k such that k concatenated with k+1 is prime.

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A136414 Put the natural numbers together without spaces and read them two at a time advancing one space each time.

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A248378 a(2n) = concatenation of n+1 with n+2, a(2n+1) = concatenation of n+2 with n+1.

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A116283 k times k+7 gives the concatenation of two numbers m and m-1.

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