A030457 -id:A030457 - 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

A341715 a(n) = smallest prime of the form n||n+1||n+2||...||n+k, where || denotes decimal concatenation, or -1 if no such prime exists.

Original entry on oeis.org

2, 3, 4567, 5, 67, 7, 89

Offset: 2

N. J. A. Sloane, Feb 21 2021

a(1) is unknown, but is believed to exist (see A007908). The corresponding value of k, if it exists, is known to be at least 300000, so in any case this prime would be too large to include in an OEIS entry, which is why this sequence has offset 2.
a(9) = 9||10||...||187 (see Example section), but that is too large to show in the data field. a(A030457(n)) = A030457(n)||A030457(n)+1 and k = 1 for n > 1. If m is in A030470 but not in A030457, then a(m) = m||m+1||m+2||m+3 and k = 3. Of course a(p) = p and k = 0 for p prime. - Chai Wah Wu, Feb 22 2021
For the corresponding values of k and n+k, see A341716 and A341717.
See also A140793 = (23, 345...109, 4567, 567...17, ...), A341720, and A084559 for the variant with k >= 1, so that a(n) > n also for prime n. - M. F. Hasler, Feb 22 2021

			Starting at 12, 13, 14, 15, 17, 19, 20 we get the primes 1213, 13, 14151617, 1516171819, 17, 19, 20212223, which are all terms of this sequence.
Here is a(9) from _Chai Wah Wu_, Feb 22 2021, a 445-digit number:
910111213141516171819202122232425262728293031323334353637383940414243444546\
    47484950515253545556575859606162636465666768697071727374757677787980818\
    28384858687888990919293949596979899100101102103104105106107108109110111\
    11211311411511611711811912012112212312412512612712812913013113213313413\
    51361371381391401411421431441451461471481491501511521531541551561571581\
    59160161162163164165166167168169170171172173174175176177178179180181182\
    183184185186187
a(16) = 16||17||...||43 is prime. Also for a(10), I searched up to k <= 10000, so if it exists it will have tens of thousands of decimal digits. Some other big terms are: for n = 18, k = 3589; for n = 35, k = 568; for n = 66, k = 937; for n = 275, k = 814.  - _Chai Wah Wu_, Feb 22 2021

N. J. A. Sloane, Table of n, k, with a question-mark for unknown values
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.

Cf. A007908, A030457, A030470, A030471, A135605, A341716, A341717, A341718.
If k in the definition is allowed to be zero we get [the present sequence, A341716, A341717], but if we require k>0 we get [A140793, A341720, A084559].
See A075022 for the largest prime factor of 1||2||...||n.

Array[Block[{k = #, s = #}, While[! PrimeQ[s], k++; s = FromDigits[IntegerDigits[s]~Join~IntegerDigits[k]]]; s] &, 8, 2] (* Michael De Vlieger, Feb 22 2021 *)

A341715(n)=if(isprime(n),n,eval(concat([Str(k)|k<-[n..A084559(n)]]))) \\ M. F. Hasler, Feb 22 2021

from sympy import isprime
def A341715(n):
    m, k = n, n
    while not isprime(m):
        k += 1
        m = int(str(m)+str(k))
    return m # Chai Wah Wu, Feb 22 2021

a(n) = concatenate(n, ..., A084559(n)) or a(n) = n if n is prime. - M. F. Hasler, Feb 22 2021

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

A032662 a(n) is the least k such that k concatenated with k + n is prime.

Original entry on oeis.org

1, 2, 1, 4, 3, 12, 1, 2, 1, 2, 3, 6, 1, 6, 3, 4, 5, 4, 5, 8, 7, 2, 21, 10, 17, 2, 1, 2, 3, 4, 1, 2, 7, 14, 3, 4, 1, 2, 1, 2, 11, 6, 5, 18, 3, 4, 3, 6, 1, 2, 1, 8, 5, 4, 7, 2, 1, 4, 5, 4, 11, 2, 1, 4, 3, 12, 1, 2, 9, 2, 3, 6, 1, 8, 9, 2, 3, 6, 1, 2, 1, 2, 5, 4, 929, 6, 3, 4, 5, 12, 1, 2, 1

Offset: 0

Patrick De Geest, May 15 1998

First terms of sequences '1', A030457, A032617-A032624, continued with displacements d > 9.

			a(22) = 21: the concatenation of 21 and 21 + 22 = 43 is 2143 and this is a prime.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A032663.

tcat:= (a,b) -> a*10^(1+ilog10(b))+b:
f:= proc(n) local k;
  for k from 1 + (n mod 2) by 2 do
     if isprime(tcat(k,k+n)) then return k fi
  od;
end proc:
map(f, [$0..100]); # Robert Israel, Sep 05 2024

Edited by Robert Israel, Sep 05 2024

A068700 The concatenation of n with n-1 and n with n+1 both yield primes (twin primes).

Original entry on oeis.org

42, 78, 102, 108, 180, 192, 270, 300, 312, 330, 342, 390, 420, 522, 540, 612, 660, 822, 840, 882, 1002, 1140, 1230, 1272, 1482, 1542, 1632, 1770, 2100, 2190, 2682, 2742, 3072, 3198, 3408, 3642, 3828, 4242, 4452, 4572, 4740, 4788, 4998, 5622, 5718, 5832

Offset: 1

Amarnath Murthy, Mar 04 2002

All terms are congruent to {0, 12, 18} mod 30. - Zak Seidov, Oct 24 2014

a(n) = 2 * A102478(n). - Reinhard Zumkeller, Jun 27 2015

			42 is a member as 4241 as well as 4243 are primes.

Zak Seidov, Table of n, a(n) for n = 1..10000

Common terms of A030458 and A052089.

Intersection of A030457 and A054211; A102478.

import Data.List.Ordered (isect)
a068700 n = a068700_list !! (n-1)
a068700_list = isect a030457_list a054211_list
-- Reinhard Zumkeller, Jun 27 2015

filter:= proc(n)
local d;
d:= ilog10(n)+1;
isprime(n*10^d+n-1) and isprime(n*10^d+n+1)
end proc:
select(filter, [$1..10^5]); # Robert Israel, Oct 24 2014

d[n_]:=IntegerDigits[n]; conQ[n_]:=And@@PrimeQ[FromDigits/@{Join[d[n],d[n+1]],Join[d[n],d[n-1]]}]; Select[Range[5850],conQ[#] &] (* Jayanta Basu, May 21 2013 *)

for(n=2,200, if(isprime(n*10^ceil(log(n-1)/log(10))+n-1)*isprime(n*10^ceil(log(n+1)/log(10))+n+1)==1,print1(n,",")))

More terms from Benoit Cloitre, Mar 09 2002

A275319 Numbers n such that n concatenated with n+1 is not a prime.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

Vincenzo Librandi, Jul 23 2016

Complement of A030457.

Conjecture: a(n) ~ n. - Charles R Greathouse IV, Jul 23 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A030457.

[n: n in [1..500] | not IsPrime(Seqint(Intseq(n+1) cat Intseq(n)))]

Select[Range[100], !PrimeQ[FromDigits[Join[IntegerDigits[#], IntegerDigits[# + 1]]]] &]

is(n)=!isprime(eval(Str(n,n+1))) \\ Charles R Greathouse IV, Jul 23 2016

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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A341715 a(n) = smallest prime of the form n||n+1||n+2||...||n+k, where || denotes decimal concatenation, or -1 if no such prime exists.

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

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A032662 a(n) is the least k such that k concatenated with k + n is prime.

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A068700 The concatenation of n with n-1 and n with n+1 both yield primes (twin primes).

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A275319 Numbers n such that n concatenated with n+1 is not a prime.

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

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