A052088 -id:A052088 - cp's OEIS frontend

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

A052089 Primes formed by concatenating k with k-1.

Original entry on oeis.org

43, 109, 2221, 2423, 3433, 4241, 5857, 7069, 7877, 8887, 10099, 102101, 108107, 112111, 114113, 124123, 148147, 154153, 160159, 172171, 180179, 192191, 198197, 202201, 208207, 210209, 214213, 238237, 244243, 262261, 264263, 268267, 270269, 282281, 294293, 300299

Offset: 1

Patrick De Geest, Jan 15 2000

			2423 is a prime and a concatenation of 24 and 23.

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

Cf. A030458, A052087, A052088.

[Seqint(Intseq(n-1) cat Intseq(n)): n in [2..300 by 2] | IsPrime(Seqint(Intseq(n-1) cat Intseq(n)))]; // Marius A. Burtea, Mar 21 2019

Sort[Select[FromDigits[Flatten[IntegerDigits/@#]]&/@Partition[ Range[ 300,1,-1],2,1],PrimeQ]] (* Harvey P. Dale, May 09 2012 *)
Select[Table[n 10^IntegerLength[n-1]+n-1,{n,2,300}],PrimeQ] (* Harvey P. Dale, Aug 20 2025 *)

for(n=4,1e4,if(isprime(t=eval(Str(n,n-1))),print1(t", "))) \\ Charles R Greathouse IV, May 07 2013

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(), 36))) # 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

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

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

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.

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.

A084551 Primes which are a concatenation of five consecutive numbers.

Original entry on oeis.org

1516171819, 3940414243, 5758596061, 6566676869, 7778798081, 8384858687, 8990919293, 129130131132133, 153154155156157, 197198199200201, 213214215216217, 239240241242243, 269270271272273, 387388389390391, 399400401402403, 443444445446447, 459460461462463

Offset: 1

Zak Seidov, Jun 27 2003

Harvey P. Dale, Table of n, a(n) for n = 1..1000

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

Select[Table[FromDigits[Flatten[IntegerDigits[n+Range[-2,2]]]],{n,2,500}],PrimeQ] (* Jayanta Basu, May 24 2013 *)
Select[FromDigits[Flatten[IntegerDigits[#]]]&/@Partition[Range[600],5,1], PrimeQ] (* Harvey P. Dale, Nov 11 2014 *)

A341931 a(n) = smallest 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, 3, -1, -1, 7, 11, -1, 13, -1, -1, -1, 17, -1, 19, -1, -1, 19, 23, 23, 13, -1, 23, -1, 29, -1, 31, -1, -1, 33, -1, -1, 37, -1, -1, -1, 41, 41, 43, -1, -1, 3, 47, 17, -1, -1, 47, 37, 41, -1, 27, 47, -1, 57, 59, 47, 61, -1, -1, -1, -1, -1

Offset: 0

Chai Wah Wu, Feb 23 2021

a(n) <= A341701(n). a(82) = 1, are there any other n such that a(n) = 1?
Primes p such that a(p) < p: 7, 53, 73, 79, 89, 103, ...
n such that a(n) < A341701(n): 7, 10, 22, 46, 48, 53, 55, 73, ...
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(7) = 3 since 76543 is prime and 765432, 7654321 are not. a(10) = 7 since 10987 is prime.

Cf. A052088, A052089, A341701, A341702, A341715, A341716, A341717, A341932.

from sympy import isprime
def A341931(n):
    k, m, r = n, n-1, n if isprime(n) else -1
    while m > 0:
        k = int(str(k)+str(m))
        if isprime(k):
            r = m
        m -= 1
    return r

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A030458 Primes formed by concatenating n with n+1.

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A052089 Primes formed by concatenating k with k-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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A030457 Numbers k such that k concatenated with k+1 is prime.

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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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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.

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