A248378 -id:A248378 - 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

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

A154530 Primes that are a concatenation of 2k and 2k+1 or 2k and 2k-1 for some k.

Original entry on oeis.org

23, 43, 67, 89, 109, 1213, 2221, 2423, 3433, 3637, 4241, 4243, 5051, 5657, 5857, 6263, 6869, 7069, 7877, 7879, 8081, 8887, 9091, 9293, 9697, 10099, 102101, 102103, 108107, 108109, 112111, 114113, 120121, 124123, 126127, 138139, 148147, 150151

Offset: 1

Pierre CAMI, Jan 11 2009

			2*1=2, 2*1+1=3, and 23 the concatenation of 2 and 3 is prime, so a(1)=23

Pierre CAMI, Table of n, a(n) for n=1..10153

Cf. A133986, A154531, A328903.

Cf. A010051, subsequence of A248378.

a154530 n = a154530_list !! (n-1)
a154530_list = filter ((== 1) . a010051') a248378_list
-- Reinhard Zumkeller, Jun 27 2015

Edited by Charles R Greathouse IV, Apr 28 2010

A371144 The smallest number such that the concatenation of n, a(n), n+1 is divisible by the concatenation of n and n+1.

Original entry on oeis.org

3, 5, 7, 0, 37, 76, 48, 98, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 560, 571, 582, 593, 604, 615, 626, 637, 648, 361, 670, 681, 692, 703, 714, 725, 736, 747, 758, 769, 780

Offset: 1

Scott R. Shannon, Mar 12 2024

For n > 10, when n starts with the digits 1, 2, 3, or 4, then a(n) = 2*n + 1. When n starts with the digits 5, 6, 7, or 8, then a(n) = 11*n + 10 for the vast majority of terms, although some outliers exist e.g., a(749) = 2251. When n starts with the digit 9, the values are somewhat more varied.
The maximum possible value for any term is the concatenation of n+1 and n, see the example for a(6) below. However except for a(6) and a(8), for the terms studied this only occurs four times for every order of magnitude increase in n, namely the four numbers consisting of all 9's except for the final digit of 0, 2, 6, or 8.
The first duplicate term is a(5) = a(18) = 37. There are 234 duplicates in the first 10000 terms.

			a(1) != 1 as "1"+"1"+"2" = 112 is not divisible by "1"+"2" = 12.
a(1) != 2 as "1"+"1"+"2" = 122 is not divisible by "1"+"2" = 12.
a(1) = 3 as "1"+"3"+"2" = 132 is divisible by "1"+"2" = 12.
a(5) = 37 as "5"+"37"+"6" = 5376 is divisible by "5"+"6" = 56.
a(6) = 76 as "6"+"76"+"7" = 6767 is divisible by "6"+"7" = 67. This is the first time the maximum possible value is required.

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Cf. A001704, A215027, A248378.

A275238 a(n) = n*(10^floor(log_10(n)+1) + 1) + (-1)^n.

Original entry on oeis.org

1, 10, 23, 32, 45, 54, 67, 76, 89, 98, 1011, 1110, 1213, 1312, 1415, 1514, 1617, 1716, 1819, 1918, 2021, 2120, 2223, 2322, 2425, 2524, 2627, 2726, 2829, 2928, 3031, 3130, 3233, 3332, 3435, 3534, 3637, 3736, 3839, 3938, 4041, 4140, 4243, 4342, 4445, 4544, 4647, 4746, 4849, 4948, 5051, 5150, 5253, 5352, 5455, 5554

Offset: 0

Ilya Gutkovskiy, Jul 21 2016

Concatenation of n with n+(-1)^n (A004442).
Subsequence of A248378.
Primes in this sequence: 23, 67, 89, 1213, 3637, 4243, 5051, 5657, 6263, 6869, 7879, 8081, 9091, 9293, 9697, 102103, ... (A030458).
Numbers n such that a(n) is prime: 2, 6, 8, 12, 36, 42, 50, 56, 62, 68, 78, 80, 90, 92, 96, 102, 108, 120, 126, 138, ... (A030457).

			a(0) =  0 + 1 = 1;
a(1) = 11 - 1 = 10;
a(2) = 22 + 1 = 23;
a(3) = 33 - 1 = 32;
a(4) = 44 + 1 = 45;
a(5) = 55 - 1 = 54, etc.
or
a(0) =  1 -> concatenation of 0 with 0 + (-1)^0 = 1;
a(1) = 10 -> concatenation of 1 with 1 + (-1)^1 = 0;
a(2) = 23 -> concatenation of 2 with 2 + (-1)^2 = 3;
a(3) = 32 -> concatenation of 3 with 3 + (-1)^3 = 2;
a(4) = 45 -> concatenation of 4 with 4 + (-1)^4 = 5;
a(5) = 54 -> concatenation of 5 with 5 + (-1)^5 = 4, etc.
........................................................
a(2k) = 1, 23, 45, 67, 89, 1011, 1213, 1415, 1617, 1819, ...

Cf. A001704, A002285, A004442, A020338, A030457, A030458, A030656, A033999, A064834, A248378.

Table[n (10^Floor[Log[10, n] + 1] + 1) + (-1)^n, {n, 0, 55}]

a(n) = if(n, n*(10^(logint(n,10)+1) + 1) + (-1)^n, 1) \\ Charles R Greathouse IV, Jul 21 2016

a(n) = A020338(n) + A033999(n).
a(2k) = A030656(k).
A064834(a(n)) > 0, for n > 0.
a(n) ~ 10*n*10^floor(c*log(n)), where c = 1/log(10) = 0.4342944819... = A002285.

cp's OEIS Frontend

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

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

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A154530 Primes that are a concatenation of 2*k and 2*k+1 or 2*k and 2*k-1 for some k.

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A371144 The smallest number such that the concatenation of n, a(n), n+1 is divisible by the concatenation of n and n+1.

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A275238 a(n) = n*(10^floor(log_10(n)+1) + 1) + (-1)^n.

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A154530 Primes that are a concatenation of 2k and 2k+1 or 2k and 2k-1 for some k.