A052089 -id:A052089 - cp's OEIS frontend

A068699 Duplicate of A052089.

43, 109, 2221, 2423, 3433, 4241, 5857, 7069, 7877, 8887, 10099, 102101, 108107

Offset: 1

dead

A030458 Primes formed by concatenating n with n+1.

23, 67, 89, 1213, 3637, 4243, 5051, 5657, 6263, 6869, 7879, 8081, 9091, 9293, 9697, 102103, 108109, 120121, 126127, 138139, 150151, 156157, 180181, 186187, 188189, 192193, 200201, 216217, 242243, 246247, 252253, 270271, 276277, 278279, 300301, 308309, 312313, 318319

Offset: 1

Patrick De Geest

Primes in A030656.

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A052089, A052087, A052088.

[m: n in [2..270 by 2] | IsPrime(m) where m is Seqint(Intseq(n+1) cat Intseq(n))];  // Bruno Berselli, Jun 18 2011

Select[Table[FromDigits[Join[Flatten[IntegerDigits[{n,n+1}]]]],{n,270}],PrimeQ] (* Jayanta Basu, May 16 2013 *)

forstep(n=2,1e3,2,if(isprime(k=eval(Str(n,n+1))),print1(k", "))) \\ Charles R Greathouse IV, Jun 18 2011

from sympy import isprime
from itertools import count, islice
def agen(): yield from filter(isprime, (int(str(k)+str(k+1)) for k in count(2, 2)))
print(list(islice(agen(), 38))) # Michael S. Branicky, Aug 05 2022

A052087 Primes whose decimal expansion is a concatenation of two or more consecutive increasing numbers (no leading zeros allowed).

Original entry on oeis.org

23, 67, 89, 1213, 3637, 4243, 4567, 5051, 5657, 6263, 6869, 7879, 8081, 9091, 9293, 9697, 102103, 108109, 120121, 126127, 138139, 150151, 156157, 180181, 186187, 188189, 192193, 200201, 216217, 242243, 246247, 252253, 270271

Offset: 1

Patrick De Geest, Jan 15 2000

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A052088, A030458, A052089.

Primes in A035333.

With[{e = 6}, TakeWhile[Select[Function[s, Union@ Flatten@ Map[Function[k, Map[FromDigits@ Flatten@ IntegerDigits@ # &, Partition[Take[s, 10^(e - k)], k, 1]]], Range[2, e]]]@ Range[10^e], PrimeQ], IntegerLength@ # <= e &]] (* Michael De Vlieger, Apr 23 2017 *)

Initial data corrected by Paul Tek, May 07 2013

A052088 Primes whose decimal expansion is a concatenation of two or more consecutive decreasing numbers (no leading zeros allowed).

Original entry on oeis.org

43, 109, 2221, 2423, 3433, 4241, 5857, 7069, 7877, 8887, 10099, 10987, 76543, 102101, 108107, 112111, 114113, 124123, 148147, 154153, 160159, 172171, 180179, 192191, 198197, 202201, 208207, 210209, 214213, 238237, 244243, 262261

Offset: 1

Patrick De Geest, Jan 15 2000

Paul Tek, Table of n, a(n) for n = 1..10000

Cf. A052087, A030458, A052089.

Initial data corrected by Paul Tek, May 07 2013

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

A210511 Primes formed by concatenating k, k, and 1 for k >= 1.

Original entry on oeis.org

331, 661, 881, 991, 18181, 20201, 21211, 26261, 27271, 32321, 33331, 41411, 48481, 51511, 54541, 57571, 60601, 65651, 69691, 71711, 78781, 86861, 89891, 90901, 92921, 98981, 99991, 1041041, 1051051, 1131131, 1191191, 1201201, 1221221, 1231231, 1261261, 1281281

Offset: 1

Abhiram R Devesh, Jan 26 2013

This sequence is similar to A030458 and A052089.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A030458, A052089.

[nn1: n in [1..130] | IsPrime(nn1) where nn1 is Seqint([1] cat Intseq(n) cat Intseq(n))]; // Bruno Berselli, Jan 30 2013

Select[Table[FromDigits[Flatten[{IntegerDigits[n], IntegerDigits[n], {1}}]], {n, 100}], PrimeQ] (* Alonso del Arte, Jan 27 2013 *)
With[{nn=200},Select[FromDigits[Flatten[IntegerDigits[#]]]&/@Thread[ {Range[ nn],Range[nn],1}],PrimeQ]] (* Harvey P. Dale, Aug 17 2013 *)

import numpy as np
def factors(n):
    return reduce(list._add_, ([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0))
for i in range(1,2000):
    p1=int(str(i)+str(i)+"1")
    if len(factors(p1))<3:
        print(p1)

from sympy import isprime
from itertools import count, islice
def agen(): yield from filter(isprime, (int(str(k)+str(k)+'1') for k in count(1)))
print(list(islice(agen(), 36))) # Michael S. Branicky, Jul 26 2022

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

A084559 Smallest a(n) > n such that concatenation of numbers from n to a(n) is a prime or 0 if no such number exists.

Original entry on oeis.org

3, 19, 7, 17, 7, 13, 9, 187

Offset: 2

Zak Seidov, Jun 27 2003

Terms a(1) and a(10) (and many other terms) are currently unknown.
a(11) = 309, a(12) = 13.
a(1) > 344869 (see A007908). - Sean A. Irvine, Jun 17 2019
More terms: a(14..17) = (17, 19, 43, 39), a(20) = 23, a(23) = 41, a(25) = 49, a(26) = 147, a(28) = 73, a(33..39) = (103, 37, 603, 37, 43, 57, 43), a(42) = 43, a(44) = 51, a(49) = 241, a(50) = 51, a(n) > 1000 for 12 < n < 50 not mentioned here. - M. F. Hasler, Feb 22 2021
a(10) > 10010, a(18) = 3607, a(66) = 1003, a(275) = 1089. If n == 2 (mod 3), then a(n) == 3 or 5 (mod 6). If n == 0 or 1 (mod 3), then a(n) == 1 (mod 6) (see A341716). - Chai Wah Wu, Feb 22 2021
a(10) > 50000. - Michael S. Branicky, Feb 25 2025

			a(4) = 7 because 4567 is a prime.

Eric Weisstein's World of Mathematics, Consecutive Number Sequences.
Eric Weisstein's World of Mathematics, Smarandache Prime

Cf. A007908, A030458, A030471, A052087, A052088, A052089.

Cf. also A341715, A341716, A341717 (similar but a(n) = n when n is prime).

A084559(n,N=n,T=10^logint(n,10))=while(T*=10,for(m=n+1,n=T-1,ispseudoprime(N=N*T+m)&&return(m))) \\ M. F. Hasler, Feb 22 2021

Edited by Max Alekseyev, Jan 28 2012
a(4) corrected by Daniel Suteu, Jun 16 2019
Escape clause added to definition by Chai Wah Wu, Feb 22 2021

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

A341701 a(n) = largest m > 0 such that the decimal concatenation n||n-1||n-2||...||m is prime, or -1 if no such prime exists.

Original entry on oeis.org

-1, -1, 2, 3, 3, 5, -1, 7, -1, -1, 9, 11, -1, 13, -1, -1, -1, 17, -1, 19, -1, -1, 21, 23, 23, 13, -1, 23, -1, 29, -1, 31, -1, -1, 33, -1, -1, 37, -1, -1, -1, 41, 41, 43, -1, -1, 39, 47, 41, -1, -1, 47, 37, 53, -1, 43, 47, -1, 57, 59, 47, 61, -1, -1, -1, -1, -1

Offset: 0

Chai Wah Wu, Feb 23 2021

A variant of A341717. a(82) = 1. Are there other n such that a(n) = 1?

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

			a(4) = 3 since 43 is prime, a(25) = 13 since 25242322212019181716151413 is prime.

Cf. A052088, A052089, A341702, A341715, A341716, A341717.

from sympy import isprime
def A341701(n):
    k, m = n, n-1
    while not isprime(k) and m > 0:
        k = int(str(k)+str(m))
        m -= 1
    return m+1 if isprime(k) else -1

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

cp's OEIS Frontend

A068699 Duplicate of A052089.

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A030458 Primes formed by concatenating n with n+1.

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A052087 Primes whose decimal expansion is a concatenation of two or more consecutive increasing numbers (no leading zeros allowed).

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A052088 Primes whose decimal expansion is a concatenation of two or more consecutive decreasing numbers (no leading zeros allowed).

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

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A210511 Primes formed by concatenating k, k, and 1 for k >= 1.

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

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A084559 Smallest a(n) > n such that concatenation of numbers from n to a(n) is a prime or 0 if no such number exists.

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