A105184 -id:A105184 - cp's OEIS frontend

A019549 Primes formed by concatenating other primes.

23, 37, 53, 73, 113, 137, 173, 193, 197, 211, 223, 227, 229, 233, 241, 257, 271, 277, 283, 293, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 433, 523, 541, 547, 557, 571, 577, 593, 613, 617, 673, 677, 719, 727, 733, 743, 757, 761, 773, 797, 977

Offset: 1

R. Muller

			113 is member as 11 and 3 are primes.
a(12)=227 = "2"+"2"+"7" is the first term not in A105184 (restricted to concatenation of two primes). [_M. F. Hasler_, Oct 15 2009]

M. F. Hasler, Table of n, a(n) for n = 1..17495
Sylvester Smith, A Set of Conjectures on Smarandache Sequences, Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.

Cf. A105184, A152242.

is_A019549(n, recurse=0)={ isprime(n) == recurse & return(recurse); for(i=1, #Str(n)-1, isprime( n%10^i ) & is_A019549( n\10^i, 1) & n\10^(i-1)%10 & return(1)) } \\ M. F. Hasler, Oct 15 2009

from sympy import isprime
def c(n, m):
    if m > 0 and isprime(n): return True
    s = str(n)
    return any(s[i]!="0" and isprime(int(s[:i])) and c(int(s[i:]), m+1) for i in range(1, len(s)))
def ok(n): return isprime(n) and c(n, 0)
print([k for k in range(978) if ok(k)]) # Michael S. Branicky, Sep 01 2024

A152242 Integers formed by concatenating primes.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 22, 23, 25, 27, 29, 31, 32, 33, 35, 37, 41, 43, 47, 52, 53, 55, 57, 59, 61, 67, 71, 72, 73, 75, 77, 79, 83, 89, 97, 101, 103, 107, 109, 112, 113, 115, 117, 127, 131, 132, 133, 135, 137, 139, 149, 151, 157, 163, 167, 172, 173, 175, 177, 179

Offset: 1

Eric Angelini, Oct 15 2009

Leading zeros are not allowed (cf. A166504).

For any k > 0, there are A246806(k) terms with k digits. - Rémy Sigrist, Jan 08 2023

			101 is a member since it is prime; 303 is not since it is composite and 30 is also not a prime.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Henri Picciotto, Selected Integer Sequences

Cf. A019549, A038692, A066737, A105184, A129800, A166504, A246806.

is_A152242(n)=/* If n is even, the last digit must be 2 and [n\10] (if nonzero) must be in this sequence. (This check is not necessary but improves speed.) */ bittest(n,0) || return( n%10==2 && (n<10 || is_A152242(n\10))); isprime(n) && return(1); for(i=1,#Str(n)-1, n%10^i>10^(i-1) && isprime( n%10^i ) && is_A152242( n\10^i) && return(1)) \\ M. F. Hasler, Oct 15 2009; edited Oct 16 2009, to disallow leading zeros

from sympy import isprime
def ok(n):
    if isprime(n): return True
    s = str(n)
    return any(s[i]!="0" and isprime(int(s[:i])) and ok(int(s[i:])) for i in range(1, len(s)))
print([k for k in range(180) if ok(k)]) # Michael S. Branicky, Sep 01 2024

More terms from M. F. Hasler and Zak Seidov, Oct 15 2009

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

A121609 Composite numbers that can be written as concatenation of two primes in decimal representation.

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, 231, 232, 235, 237, 243, 247, 253, 259, 261, 267, 273, 279, 289, 292, 295, 297, 312, 315, 319, 323, 329, 341, 343, 361, 371, 372, 375, 377

Offset: 1

Reinhard Zumkeller, Aug 10 2006

Subsequence of A066737.

			A002808(249) = 315 = 31*10+5 = A000040(11)*10+A000040(3),
therefore 315 is a term: a(44) = 315;
A002808(252) = 319 = 3*100+19 = A000040(2)*100+A000040(8),
therefore 319 is a term: a(45) = 319.

Jean-Marc Falcoz, Table of n, a(n) for n = 1..11140

Cf. A121610, A105184, A121608.

A380943 Primes written in decimal representation by the concatenation of primes p and q such that the concatenation of q and p also forms a prime.

Original entry on oeis.org

37, 73, 113, 173, 197, 311, 313, 317, 331, 337, 359, 367, 373, 593, 617, 673, 719, 733, 761, 797, 977, 1093, 1097, 1123, 1277, 1319, 1361, 1373, 1783, 1913, 1931, 1979, 1997, 2293, 2297, 2311, 2347, 2389, 2713, 2837, 2971, 3109, 3119, 3137, 3191, 3229, 3271

Offset: 1

James C. McMahon, Apr 03 2025

Member primes could be named Saint Patrick primes because the date of Saint Patrick's Day, March 17 (3/17), produces the terms 173 and 317.

The primes 37, 73, and 373 uniquely satisfy the requirements if cleaved at every possible position. - Robert G. Wilson v, May 03 2025

			Primes 173 and 317 are members because they are formed by the concatenation of 17 & 3  and 3 & 17, respectively.
While the concatenation of 13 and 7 forms the prime 137, it is not a member because the concatenation of 7 and 13 is 713, which is not prime.

James C. McMahon, Table of n, a(n) for n = 1..1000
Michael De Vlieger, Plot a(n) = p<>q at (x,y) = (pi(p), pi(q)) showing all terms with pi(p) <= 40 and pi(q) <= 40, labeling p in red and q in blue.
Michael De Vlieger, Plot a(n) = p<>q at (x,y) = (pi(p), pi(q)) showing all terms with pi(p) <= 2000 and pi(q) <= 2000.

Subsequence of A019549 and A105184.

Cf. A133187.

lim=3300; plim=Max[FromDigits[Rest[IntegerDigits[lim]]], FromDigits[Drop[IntegerDigits[lim], -1]]]; p=Prime[Range[PrimePi[plim]]];p2=Subsets[p,{2}];fp[{a_,b_}]:=FromDigits[Join[IntegerDigits[a],IntegerDigits[b]]];rfp[{a_,b_}]:=FromDigits[Join[IntegerDigits[b],IntegerDigits[a]]];pabba=Select[p2,PrimeQ[fp[#]]&&PrimeQ[rfp[#]]&];pp1=fp/@pabba;pp2=rfp/@pabba;Select[Union[Join[pp1,pp2]],#<=lim&]

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:])) and isprime(int(s[i:]+s[:i])) for i in range(1, len(s)))
print([k for k in range(3300) if ok(k)]) # Michael S. Branicky, Apr 05 2025

A238056 Primes which are the concatenation of two primes in exactly one way.

Original entry on oeis.org

23, 37, 53, 73, 113, 137, 173, 193, 197, 211, 223, 229, 233, 241, 271, 283, 293, 311, 331, 337, 347, 353, 359, 367, 379, 383, 389, 397, 433, 523, 541, 547, 571, 593, 613, 617, 673, 677, 719, 733, 743, 761, 773, 977, 1013, 1033, 1093, 1097, 1117, 1123, 1129

Offset: 1

Colin Barker, Feb 17 2014

This is not a duplicate of A129800, which accepts "07" for example as the second prime.

			113 is in the sequence because 11 and 3 are both primes, but 1 and 13 are not both primes, so there is one way.

Giovanni Resta, Table of n, a(n) for n = 1..5000

Cf. A105184, A238057, A129800.

Cf. A010051, A000040.

a238056 n = a238056_list !! (n-1)
a238056_list = filter ((== 1) . length . f) a000040_list where
  f x = filter (\(us, vs) ->
               head vs /= '0' &&
               a010051' (read us :: Integer) == 1 &&
               a010051' (read vs :: Integer) == 1) $
               map (flip splitAt $ show x) [1 .. length (show x) - 1]
-- Reinhard Zumkeller, Feb 27 2014

spl[n_] := Block[{d = IntegerDigits@n, c = 0, z}, z = Length@d; Do[If[PrimeQ@ FromDigits@ Take[d, k] && d[[k + 1]] > 0 && PrimeQ@ FromDigits@ Take[d, k - z], c++], {k, z - 1}]; c]; Select[ Prime@ Range@ 300, spl[#] == 1 &] (* Giovanni Resta, Feb 27 2014 *)

A238057 Primes which are the concatenation of two primes in exactly two ways.

Original entry on oeis.org

313, 317, 373, 797, 1373, 1913, 1973, 1997, 2113, 2293, 2311, 2347, 2383, 2389, 2953, 2971, 3167, 3313, 3373, 3593, 3673, 3677, 3719, 3733, 3761, 4337, 4397, 5233, 5347, 5953, 6173, 6197, 6737, 7193, 7331, 7433, 7577, 7877, 7919, 7937, 10313, 10337, 10937

Offset: 1

Colin Barker, Feb 17 2014

			313 is in the sequence because 31 and 3 are both primes, and 3 and 13 are both primes, so there are two ways.

Giovanni Resta, Table of n, a(n) for n = 1..1000

Cf. A105184, A238056.

Cf. A010051, A000040.

a238057 n = a238057_list !! (n-1)
a238057_list = filter ((== 2) . length . f) a000040_list where
  f x = filter (\(us, vs) ->
               head vs /= '0' &&
               a010051' (read us :: Integer) == 1 &&
               a010051' (read vs :: Integer) == 1) $
               map (flip splitAt $ show x) [1 .. length (show x) - 1]
-- Reinhard Zumkeller, Feb 27 2014

spl[n_] := Block[{d = IntegerDigits@n, c = 0, z}, z = Length@d; Do[ If[ PrimeQ@ FromDigits@ Take[d, k] && d[[k + 1]] > 0 && PrimeQ@ FromDigits@ Take[d, k - z], c++], {k, z - 1}]; c]; Select[ Prime@ Range@1400, spl[#] == 2 &] (* Giovanni Resta, Feb 27 2014 *)

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

A383815 Palindromic primes in A380943.

Original entry on oeis.org

313, 373, 797, 11311, 13331, 13931, 17971, 19991, 31013, 35353, 36263, 36563, 38783, 71317, 79397, 97379, 98389, 1129211, 1196911, 1611161, 1793971, 1982891, 3106013, 3166613, 3193913, 3236323, 3288823, 3304033, 3319133, 3329233, 3365633, 3417143, 3447443, 3449443, 3515153, 3670763

Offset: 1

James C. McMahon and Robert G. Wilson v, Jun 06 2025

A380943 requires that primes, p_n, written in decimal representation by the concatenation of primes p and q such that the concatenation of q and p also forms a prime.

Intersection of A002385 and A380943.

			The palindromic prime 313 is formed by the concatenation of the primes 31 and 3, which reversed, also form the prime 331. The palindromic prime 13931 is formed by the concatenation of 139 and 31; 31139 is also prime.

Cf. A000040, A002385, A105184, A380943, A383810, A383811, A383812, A383813.

rev:= proc(n) local L,i;
   L:= convert(n,base,10);
   add(L[-i]*10^(i-1),i=1..nops(L))
end proc:
tcat:= proc(x,y) y + 10^(ilog10(y)+1)*x end proc:
filter:= proc(z) local i,x,y;
  if not isprime(z) then return false fi;
  for i from 1 to ilog10(z) do
    x:= z mod 10^i;
    if x < 10^(i-1) then next fi;
    y:= (z-x)/10^i;
    if isprime(x) and isprime(y) and isprime(tcat(x,y)) then return true fi;
  od;
  false
end proc:
N:= 7: # for terms of up to 7 digits
R:= NULL:
for d from 1 to (N-1)/2 do
  for x from 10^(d-1) to 10^d-1 do
    for y from 0 to 9 do
      z:= rev(x) + 10^d * y + 10^(d+1)*x;
      if filter(z) then R:= R,z fi
od od od:
R;  # Robert Israel, Jun 08 2025

f[n_] := Block[{cnt = 0, id = IntegerDigits@ n, k = 1, len, p, q, qp}, len = Length@ id; While[k < len, p = Take[id, k]; q = Take[id, -len + k]; qp = FromDigits[Join[q, p]]; If[ PrimeQ@ FromDigits@ p && PrimeQ@ FromDigits@ q && PrimeQ@ qp && IntegerLength@ qp == len, cnt++]; k++]; cnt]; fQ[n_] := Reverse[idn = IntegerDigits@ n] == idn && f@ n > 0; Select[ Prime@ Range@ 264000, fQ]

A383816 Palindromic primes which satisfy the requirements of A380943 in at least two ways.

Original entry on oeis.org

373, 1793971, 7933397, 374636473, 714707417, 727939727, 787333787, 790585097, 947939749, 991999199, 10253935201, 11365556311, 11932823911, 13127372131, 34390609343, 35369996353, 35381318353, 36297179263, 37018281073, 37423332473, 37773537773, 38233333283, 38914541983, 39064546093

Offset: 1

James C. McMahon and Robert G. Wilson v, Jun 09 2025

Terms of A380943 are primes whose decimal representation is the concatenation of primes p and q such that the concatenation of q and p also forms a prime.

			The palindromic prime 373 meets the requirements of A380943 in two ways: the concatenation of 3 and 37 forms the prime 337, and the concatenation of 73 and 3 forms the prime 733.
Although 37673 is a palindrome where 3, 7673, and 76733 are all primes and 3767, 3, and 33767 are all primes, the palindrome is not prime and is therefore not in the sequence.

Subsequence of A383810.

Cf. A000040, A002385, A105184, A380943.

f[n_] := Block[{cnt = 0, id = IntegerDigits@ n, k = 1, len, p, q, qp}, len = Length@ id; While[k < len, p = Take[id, k]; q = Take[id, -len + k]; qp = FromDigits[Join[q, p]]; If[ PrimeQ@ FromDigits@ p && PrimeQ@ FromDigits@ q && PrimeQ@ qp && IntegerLength@ qp == len, cnt++]; k++]; cnt]; fQ[n_] := Reverse[idn = IntegerDigits@ n] == idn && f@ n > 1; Select[ Prime@ Range@ 3000000, fQ]

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A019549 Primes formed by concatenating other primes.

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A152242 Integers formed by concatenating primes.

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A106582 Numbers which are the concatenation of two primes.

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A121609 Composite numbers that can be written as concatenation of two primes in decimal representation.

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A380943 Primes written in decimal representation by the concatenation of primes p and q such that the concatenation of q and p also forms a prime.

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A238056 Primes which are the concatenation of two primes in exactly one way.

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A238057 Primes which are the concatenation of two primes in exactly two ways.

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

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