A054211 -id:A054211 - cp's OEIS frontend

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

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

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

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

A341702 a(n) is the smallest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

Original entry on oeis.org

-1, -1, 0, 0, 1, 0, -1, 0, -1, -1, 1, 0, -1, 0, -1, -1, -1, 0, -1, 0, -1, -1, 1, 0, 1, 12, -1, 4, -1, 0, -1, 0, -1, -1, 1, -1, -1, 0, -1, -1, -1, 0, 1, 0, -1, -1, 7, 0, 7, -1, -1, 4, 15, 0, -1, 12, 9, -1, 1, 0, 13, 0, -1, -1, -1, -1, -1, 0, 57, -1, 1, 0, -1, 0

Offset: 0

Chai Wah Wu, Feb 23 2021

A variation of A341716. a(n) = n-1 for n = 82. Are there other n such that a(n) = n-1?

Similar argument as in A341716 shows that if n > 3 and a(n) >= 0, then n-a(n) is odd, a(n) !== 2 (mod 3) and 2n-a(n) !== 0 (mod 3).

			a(10) = 1 since 109 is prime. a(22) = 1 since 2221 is prime.

Robert Israel, Table of n, a(n) for n = 0..2000

Cf. A052088, A052089, A054211, A341701, A341715, A341716, A341717.

tcat:= proc(x,y) x*10^(1+ilog10(y))+y end proc:
f:= proc(n) local x,k;
  x:= n;
  for k from 0 to n-1 do
    if isprime(x) then return k fi;
    x:= tcat(x,n-k-1)
  od;
  -1
end proc:
map(f, [$0..100]); # Robert Israel, Mar 02 2022

a(n) = my(k=0, s=Str(n)); while (!isprime(eval(s)), k++; n--; if (k>=n, return(-1)); s = concat(s, Str(n-k))); return(k); \\ Michel Marcus, Mar 02 2022

from sympy import isprime
def A341702(n):
    k, m = n, n-1
    while not isprime(k) and m > 0:
        k = int(str(k)+str(m))
        m -= 1
    return n-m-1 if isprime(k) else -1

a(n) = n-A341701(n).

a(p) = 0 if and only if p is prime.

A089432 Numbers n such that n concatenated with floor(n/2) is prime.

Original entry on oeis.org

3, 7, 15, 19, 23, 27, 35, 39, 47, 55, 67, 75, 95, 99, 107, 119, 127, 135, 163, 179, 187, 195, 227, 235, 239, 243, 255, 267, 303, 319, 327, 347, 379, 407, 427, 443, 455, 459, 463, 479, 523, 539, 547, 555, 583, 607, 619, 635, 663, 667, 683, 687, 695, 715, 723

Offset: 1

Reinhard Zumkeller, Dec 28 2003

All terms are odd.

Cf. A054211.

cp's OEIS Frontend

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

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A030457 Numbers k such that k concatenated with k+1 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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A341702 a(n) is the smallest k < n such that the decimal concatenation n||n-1||n-2||...||n-k is prime, or -1 if no such prime exists.

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Programs

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Python

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A089432 Numbers n such that n concatenated with floor(n/2) is prime.

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