A121609 -id:A121609 - cp's OEIS frontend

A105184 Primes that can be written as concatenation of two primes in decimal representation.

23, 37, 53, 73, 113, 137, 173, 193, 197, 211, 223, 229, 233, 241, 271, 283, 293, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 433, 523, 541, 547, 571, 593, 613, 617, 673, 677, 719, 733, 743, 761, 773, 797, 977, 1013, 1033, 1093

Offset: 1

Lekraj Beedassy, Apr 11 2005

Primes that can be written as the concatenation of two distinct primes is the same sequence.
Number of terms < 10^n: 0, 4, 48, 340, 2563, 19019, 147249, ... - T. D. Noe, Oct 04 2010
The second prime cannot begin with the digit zero, else 307 would be the first additional term. - Michael S. Branicky, Sep 01 2024

			193 is in the sequence because it is the concatenation of the primes 19 and 3.
197 is in the sequence because it is the concatenation of the primes 19 and 7.
199 is not in the sequence because there is no way to break it into two substrings such that both are prime: neither 1 nor 99 is prime, and 19 is prime but 9 is not.

T. D. Noe, Table of n, a(n) for n=1..10000

Subsequence of A019549.

Cf. A121608, A121609, A121610, A083427, A129800.

searchMax = 10^4; Union[Reap[Do[p = Prime[i]; q = Prime[j]; n = FromDigits[Join[IntegerDigits[p], IntegerDigits[q]]]; If[PrimeQ[n], Sow[n]], {i, PrimePi[searchMax/10]}, {j, 2, PrimePi[searchMax/10^Ceiling[Log[10, Prime[i]]]]}]][[2, 1]]] (* T. D. Noe, Oct 04 2010 *)
Select[Prime@Range@1000,
 MatchQ[IntegerDigits@#, {x__, y__} /;
    PrimeQ@FromDigits@{x} && First@{y} != 0 &&
PrimeQ@FromDigits@{y}] &] (* Hans Rudolf Widmer, Nov 30 2024 *)

from sympy import isprime
def ok(n):
    if not isprime(n): return False
    s = str(n)
    return any(s[i]!="0" and isprime(int(s[:i])) and isprime(int(s[i:])) for i in range(1, len(s)))
print([k for k in range(1100) if ok(k)]) # Michael S. Branicky, Sep 01 2024

Corrected and extended by Ray Chandler, Apr 16 2005
Edited by N. J. A. Sloane, May 03 2007
Edited by N. J. A. Sloane, to remove erroneous b-file, comments and Mma program, Oct 04 2010

A106582 Numbers which are the concatenation of two primes.

Original entry on oeis.org

22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 112, 113, 115, 117, 132, 133, 135, 137, 172, 173, 175, 177, 192, 193, 195, 197, 211, 213, 217, 219, 223, 229, 231, 232, 233, 235, 237, 241, 243, 247, 253, 259, 261, 267, 271, 273, 279, 283, 289

Offset: 1

Eric Angelini and Robert G. Wilson v, May 09 2005

A105184 and A121609 are subsequences.

			133 is in the sequence because 133 = 13*10+3 = A000040(6)*10+A000040(2).

T. D. Noe, Table of n, a(n) for n = 1..10000
G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 17257

Cf. A001358, A105184, A121609.

nn=500; t=Union[Reap[Do[n=FromDigits[Join[IntegerDigits[Prime[i]], IntegerDigits[Prime[j]]]]; If[n<=nn, Sow[n]], {i,PrimePi[nn/10]}, {j,PrimePi[nn/IntegerDigits[nn][[1]]]}]][[2,1]]] (* T. D. Noe, Mar 11 2011 *)
Take[FromDigits[Flatten[IntegerDigits/@#]]&/@Tuples[Prime[Range[30]],2]//Union,60] (* Harvey P. Dale, May 28 2025 *)

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    for k in count(1):
        s = str(k)
        if any(s[i] != '0' and isprime(int(s[:i])) and isprime(int(s[i:])) for i in range(1, len(s))):
            yield k
print(list(islice(agen(), 55))) # Michael S. Branicky, Feb 26 2022

Corrected by Arkadiusz Wesolowski, Mar 11 2011

A066737 Composite numbers that are concatenations of primes.

Original entry on oeis.org

22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 112, 115, 117, 132, 133, 135, 172, 175, 177, 192, 195, 213, 217, 219, 222, 225, 231, 232, 235, 237, 243, 247, 252, 253, 255, 259, 261, 267, 272, 273, 275, 279, 289, 292, 295, 297, 312, 315, 319, 322, 323, 325

Offset: 1

Joseph L. Pe, Jan 15 2002

			72 is the concatenation of primes 7 and 2. 132 is the concatenation of primes 13 and 2. 225 is the concatenation of 2, 2 and 5.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A121609.

ccat:= proc(m,n) 10^(1+ilog10(n))*m+n end proc:
C[1]:= {2,3,5,7}: P[1]:=C[1]:
for n from 2 to 3 do
  P[n]:= select(isprime, {seq(i,i=10^(n-1)+1..10^n-1,2)});
  C[n]:= P[n];
  for m from 1 to n-1 do
    C[n]:= C[n] union {seq(seq(ccat(p,q),p =P[m]),q=C[n-m])};
  od
od:
seq(op(sort(convert(remove(isprime,C[n]),list))),n=1..3); # Robert Israel, Jan 22 2020

for(n=1,999, !isprime(n) && is_A152242(n) && print1(n", ")) \\ M. F. Hasler, Oct 16 2009

A066737 = A152242 \ A000040 = A152242 intersect A002808. - M. F. Hasler, Oct 16 2009

More terms from Larry Reeves (larryr(AT)acm.org), Apr 03 2002

Missing terms added by M. F. Hasler, Oct 16 2009

A121608 Primes that can be written as concatenation of two composite numbers in decimal representation.

Original entry on oeis.org

89, 109, 149, 229, 269, 349, 359, 389, 409, 421, 433, 439, 449, 457, 463, 487, 491, 499, 509, 569, 659, 677, 691, 709, 769, 809, 821, 827, 829, 839, 857, 859, 863, 877, 881, 887, 919, 929, 977, 991, 1009, 1021, 1033, 1039, 1049, 1051, 1063, 1069, 1087, 1091

Offset: 1

Reinhard Zumkeller, Aug 10 2006

			A000040(169) = 1009 = 100*10+9 = A002808(74)*10+A002808(4), therefore 1009 is a term: a(41) = 1009;
A000040(172) = 1021 = 10*100+21 = A002808(5)*100+A002808(12), therefore 1021 is a term: a(42) = 1021.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A105184, A121609, A121610.

from sympy import isprime
def comp(s): i=int(s); return s[0]!='0' and i > 1 and not isprime(i)
def ok(n):
  s = str(n)
  for i in range(1, len(s)):
    if comp(s[:i]) and comp(s[i:]) and isprime(int(s)): return True
print([m for m in range(1092) if ok(m)]) # Michael S. Branicky, Feb 27 2021

A121610 Composite numbers that can be written as concatenation of two composite numbers in decimal representation.

Original entry on oeis.org

44, 46, 48, 49, 64, 66, 68, 69, 84, 86, 88, 94, 96, 98, 99, 104, 106, 108, 124, 126, 128, 129, 144, 146, 148, 154, 156, 158, 159, 164, 166, 168, 169, 184, 186, 188, 189, 204, 206, 208, 209, 214, 216, 218, 219, 224, 226, 228, 244, 246, 248, 249, 254, 256, 258

Offset: 1

Reinhard Zumkeller, Aug 10 2006

			A002808(328) = 408 = 40*10+8 = A002808(27)*10+A002808(3),
therefore 408 is a term: a(99) = 408;
A002808(331) = 412 = 4*100+12 = A002808(1)*100+A002808(6),
therefore 412 is a term: a(100) = 412.

Cf. A121609, A105184, A121608.

A167459 Composite numbers in A166504, i.e., whose decimal expansion can be split up into prime numbers, with leading zeros allowed.

Original entry on oeis.org

22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75, 77, 112, 115, 117, 132, 133, 135, 172, 175, 177, 192, 195, 202, 203, 205, 207, 213, 217, 219, 222, 225, 231, 232, 235, 237, 243, 247, 252, 253, 255, 259, 261, 267, 272, 273, 275, 279, 289, 292, 295, 297, 302, 303

Offset: 1

M. F. Hasler, Nov 19 2009

In contrast to A066737 (which is a subsequence of this one), we allow for leading zeros in the "prime" substrings; the two sequences differ from n=24 on, with a(24)=202 which is not in A066737.

Sequence A166505 gives the difference, A167459 \ A066737 = A166504 \ A152242. Sequence A167458 gives the indices of the terms not in A066737.

Cf. A002808, A066737, A121609, A166504, A167505.

is_A167459(n) = !isprime(n) & is_A166504(n)

A167459 = A002808 n A166504, where "n" means intersection.

A167459 \ A066737 = A166504 \ A152242.

A278799 Prime numbers that can be written as concatenation of two nonprimes in decimal representation.

Original entry on oeis.org

11, 19, 41, 61, 89, 101, 109, 127, 139, 149, 151, 157, 163, 181, 191, 193, 199, 211, 229, 241, 251, 269, 271, 281, 331, 349, 359, 389, 401, 409, 421, 433, 439, 449, 457, 461, 463, 487, 491, 499, 509, 521, 541, 569, 571, 601, 631, 641, 659, 661, 677, 691, 701, 709, 751, 761, 769, 809

Offset: 1

Eric Angelini and Jean-Marc Falcoz, Nov 28 2016

This is not A066738 as we concatenate exactly two nonprimes here.

A121609 is the dual sequence where "prime" and "nonprime" are switched in the definition.

			11 (prime) is the concatenation of "1" (nonprime) and "1" (nonprime); the next prime term cannot be 13 as "3" is a concatenated prime; the next prime term cannot be 17 as "7" is a concatenated prime; the next prime term is 19 as "1" and "9" are both nonprimes; the next prime term cannot be less than 41 because all terms < 41 and > 19 start with either a "2" or a "3", which are primes; etc.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..7135

Cf. A066738, A121609.

is(n)=if(!isprime(n), return(0)); my(d=digits(n)); for(i=2,#d, if(d[i] && !isprime(fromdigits(d[1..i-1])) && !isprime(fromdigits(d[i..#d])), return(1))); 0 \\ Charles R Greathouse IV, Nov 28 2016

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A105184 Primes that can be written as concatenation of two primes in decimal representation.

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A106582 Numbers which are the concatenation of two primes.

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A066737 Composite numbers that are concatenations of primes.

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A121608 Primes that can be written as concatenation of two composite numbers in decimal representation.

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A121610 Composite numbers that can be written as concatenation of two composite numbers in decimal representation.

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A167459 Composite numbers in A166504, i.e., whose decimal expansion can be split up into prime numbers, with leading zeros allowed.

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A278799 Prime numbers that can be written as concatenation of two nonprimes in decimal representation.

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PARI