A127423 -id:A127423 - cp's OEIS frontend

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

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

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

A054211 Numbers k such that k concatenated with k-1 is prime.

Original entry on oeis.org

4, 10, 22, 24, 34, 42, 58, 70, 78, 88, 100, 102, 108, 112, 114, 124, 148, 154, 160, 172, 180, 192, 198, 202, 208, 210, 214, 238, 244, 262, 264, 268, 270, 282, 294, 300, 304, 312, 328, 330, 334, 340, 342, 354, 372, 384, 390, 394, 412, 414, 420, 424, 444, 454

Offset: 1

Patrick De Geest, Feb 15 2000

A010051(A127423(a(n))) = 1. - Reinhard Zumkeller, Jun 27 2015

All terms are even. - Michel Marcus, Oct 14 2016

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

Cf. A052089, A030457, A030458, A052087, A052088.

Cf. A010051, A068700 (subsequence), A127423.

a054211 n = a054211_list !! (n-1)
a054211_list = filter ((== 1) . a010051' . a127423) [1..]
-- Reinhard Zumkeller, Jun 27 2015 Jul 15 2012

ncpQ[{a_,b_}]:=PrimeQ[FromDigits[Flatten[IntegerDigits[{b,a}]]]]; Transpose[ Select[Partition[Range[500],2,1],ncpQ]][[2]] (* Harvey P. Dale, Nov 25 2012 *)
Select[Range[500],PrimeQ[#*10^IntegerLength[#-1]+#-1]&] (* Harvey P. Dale, Mar 16 2019 *)

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

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 *)

A215028 a(1) = 1; for n >= 1, a(n+1) = (concatenation of n+1 and n) - a(n).

Original entry on oeis.org

1, 20, 12, 31, 23, 42, 34, 53, 45, 64, 1046, 165, 1147, 266, 1248, 367, 1349, 468, 1450, 569, 1551, 670, 1652, 771, 1753, 872, 1854, 973, 1955, 1074, 2056, 1175, 2157, 1276, 2258, 1377, 2359, 1478, 2460, 1579, 2561, 1680, 2662, 1781, 2763, 1882, 2864, 1983

Offset: 1

N. J. A. Sloane, Aug 04 2012

A variation of A215027.

Cf. A215027, A127423.

f:=proc(i) (i+1)*10^(1+floor(evalf(log10(i))))+i; end: # A127423
a:=proc(n) option remember; global f; if n=1 then 1 else f(n-1)-a(n-1); fi; end;

A215028(n, print_all=0)={my(a=1); for(n=1, n-1, print_all & print1(a","); a=(n+1)*10^#Str(n)+n-a); a} \\ M. F. Hasler, Aug 23 2012

A256285 a(n) = smallest divisor of the concatenation of n+1 and n, that did not occur earlier.

Original entry on oeis.org

1, 2, 43, 3, 5, 4, 29, 7, 109, 6, 173, 8, 9, 757, 17, 11, 23, 14, 673, 10, 2221, 18, 2423, 631, 15, 47, 257, 12, 13, 313, 359, 28, 3433, 19, 727, 467, 1279, 22, 577, 20, 4241, 26, 1481, 16, 929, 21, 37, 1237, 27, 25, 59, 24, 41, 2777, 39, 1439, 5857, 331, 73

Offset: 1

Eric Angelini and Reinhard Zumkeller, Jun 03 2015

Is this a permutation of the integers > 0 ?
From Robert Israel, May 20 2024: (Start)
Yes, this is a permutation of the positive integers.
For any positive integer k, there are arbitrarily large d such that 10^(d-1) > k and GCD(10^d + 1, k) == 1.  For such d, there is n such that  n == -10^d (10^d + 1)^(-1) (mod k) and 10^d > n >= 10^(d-1), and this implies that the concatenation of n+1 and n, which is 10^d * (n+1) + n, is divisible by k.  After all numbers < k have occurred, the next such n must have a(n) = k. (End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Éric Angelini, Divisors of the concatenation of n+1 and n, SeqFan list, Jun 03 2015.

Cf. A027750, A127423, A253253.

import Data.List (insert); import Data.List.Ordered (minus)
a256285 n = a256285_list !! (n-1)
a256285_list = f (tail a127423_list) [] where
   f (x:xs) ds = y : f xs (insert y ds) where
                 y = head (a027750_row x `minus` ds)

R:= NULL: S:= {}:
for n from 1 to 100 do
  v:=  10^(1+ilog10(n))*(n+1)+n;
  s:= min(numtheory:-divisors(v) minus S);
  R:= R,s;
  S:= S union {s};
od:
R; # Robert Israel, May 20 2024

A277351 Value of (n+1,n) concatenated in binary representation.

Original entry on oeis.org

5, 14, 19, 44, 53, 62, 71, 152, 169, 186, 203, 220, 237, 254, 271, 560, 593, 626, 659, 692, 725, 758, 791, 824, 857, 890, 923, 956, 989, 1022, 1055, 2144, 2209, 2274, 2339, 2404, 2469, 2534, 2599, 2664, 2729, 2794, 2859, 2924, 2989, 3054, 3119, 3184, 3249, 3314

Offset: 1

Paolo P. Lava, Oct 10 2016

			Binary representation of 12 and 13 are 1100 and 1101. Then, concat(1101,1100) = 11011100 converted in decimal representation is 220.

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

Cf. A007088, A070939, A087737, A127423.

f:= n -> (n+1)*2^(1+ilog2(n))+n:
map(f, [$1..100]); # Robert Israel, Oct 14 2016

Table[FromDigits[Join @@ Map[IntegerDigits[#, 2] &, {n + 1, n}], 2], {n, 50}] (* Michael De Vlieger, Oct 14 2016 *)

a(n) = subst(Pol(concat(binary(n+1), binary(n))), x, 2); \\ Michel Marcus, Oct 10 2016

a(n) = (n+1)*2^(1+logint(n,2)) + n; \\ after Maple; Michel Marcus, Oct 15 2016

def a(n): return int(bin(n+1)[2:] + bin(n)[2:], 2)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, May 14 2021

a(n) = (n+1) * 2^A070939(n) + n.

G.f.: (1-x)^(-2)*(5*x - 2*x^2 + Sum_{m>=1} ((2^(2*m)+2^m)*x^(2^m) - 2^(2*m)*x^(2^m+1))). - Robert Israel, Oct 14 2016

A317947 Concatenate n and n-1 in binary.

Original entry on oeis.org

10, 101, 1110, 10011, 101100, 110101, 111110, 1000111, 10011000, 10101001, 10111010, 11001011, 11011100, 11101101, 11111110, 100001111, 1000110000, 1001010001, 1001110010, 1010010011, 1010110100, 1011010101, 1011110110, 1100010111, 1100111000, 1101011001, 1101111010, 1110011011, 1110111100

Offset: 1

N. J. A. Sloane, Aug 11 2018

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000523, A007088, A127423 (base 10 version).

f:= proc(n) local A;
  A:= convert(n-1,binary);
  10^length(A)*convert(n,binary)+A
end proc:
f(1):= 10:
map(f, [$1..100]); # Robert Israel, Aug 12 2018

zbinary(n) = if (n, binary(n), [0]);
a(n) = fromdigits(concat(zbinary(n), zbinary(n-1))); \\ Michel Marcus, Aug 12 2018

a(n) = A007088(n-1)+10^(1+A000523(n-1))*A007088(n) for n >= 2. - Robert Israel, Aug 12 2018

More terms from Robert Israel, Aug 12 2018

A376252 Concatenated (n+1)||n modulo n.

Original entry on oeis.org

0, 0, 1, 2, 0, 4, 3, 2, 1, 0, 1, 4, 9, 2, 10, 4, 15, 10, 5, 0, 16, 12, 8, 4, 0, 22, 19, 16, 13, 10, 7, 4, 1, 32, 30, 28, 26, 24, 22, 20, 18, 16, 14, 12, 10, 8, 6, 4, 2, 0, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20

Offset: 1

Stuart Coe, Sep 17 2024

There is an interesting and striking pattern in the graph of this sequence that appears at n >= 20 and appears to continue indefinitely.
There does not appear to be a corresponding pattern for other bases.
Beyond the first two terms, zeros only appear where n is a multiple of 5.

			For n=2: 32 mod 2 is 0.
For n=123: 124123 mod 123 is 16.

Cf. A055642, A127423.

a[n_]:=Mod[FromDigits[Join[IntegerDigits[n+1],IntegerDigits[n]]],n]; Array[a,80] (* Stefano Spezia, Sep 18 2024 *)

a(n) = eval(concat(Str(n+1), Str(n))) % n; \\ Michel Marcus, Sep 17 2024

a(n) = 10*10^logint(n, 10) % n; \\ Ruud H.G. van Tol, Oct 26 2024

def a(n): return int(str(n+1)+str(n))%n
print([a(n) for n in range(1, 81)]) # Michael S. Branicky, Sep 17 2024

def A376252(n): return int('1'+str(n))%n # Chai Wah Wu, Oct 01 2024

a(n) = 10^A055642(n) mod n. Concatenation of 1||n modulo n. - Chai Wah Wu, Oct 01 2024

cp's OEIS Frontend

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

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

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A054211 Numbers k such that k concatenated with k-1 is prime.

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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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A215028 a(1) = 1; for n >= 1, a(n+1) = (concatenation of n+1 and n) - a(n).

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A256285 a(n) = smallest divisor of the concatenation of n+1 and n, that did not occur earlier.

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Maple

A277351 Value of (n+1,n) concatenated in binary representation.

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