A083427 -id:A083427 - 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

A361530 Primes that can be written as the result of shuffling the decimal digits of two primes.

Original entry on oeis.org

23, 37, 53, 73, 113, 127, 131, 137, 139, 151, 157, 173, 179, 193, 197, 211, 223, 229, 233, 239, 241, 271, 283, 293, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 389, 397, 421, 431, 433, 457, 523, 541, 547, 571, 593, 613, 617, 631, 673, 677, 719

Offset: 1

Robert C. Lyons, Mar 14 2023

Each term is essentially an element of the shuffle product of the decimal digits of two primes (possibly equal).

			37 and 73 are in the sequence because they are both the result of shuffling 3 and 7.
127 is in the sequence because it is the result of shuffling 2 and the digits of 17.
1193 is in the sequence because it is the result of shuffling the digits of 13 and the digits of 19.
163 is not in the sequence because it is not the result of shuffling the digits of two primes. 163 is the result of permuting the digits of 3 and 61; however, 163 contains the digits of 61 in the wrong order.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
John D. Cook, Shuffle product.
Wikipedia, Shuffle product.

Cf. A019549, A083427, A105184.

import sympy
def get_shuffle_product(list_1, list_2):
    shuffle_product = set()
    shuffle = []
    _get_shuffle_product(list_1, list_2, shuffle, shuffle_product)
    return shuffle_product
def _get_shuffle_product(list_1, list_2, shuffle, shuffle_product):
    if len(list_1) == 0 and len(list_2) == 0:
        shuffle_product.add(tuple(shuffle))
        return
    else:
        if len(list_1) == 0:
            shuffle.append(list_2[0])
            _get_shuffle_product(list_1, list_2[1:], shuffle, shuffle_product)
            shuffle.pop()
        elif len(list_2) == 0:
            shuffle.append(list_1[0])
            _get_shuffle_product(list_1[1:], list_2, shuffle, shuffle_product)
            shuffle.pop()
        else:
            shuffle.append(list_1[0])
            _get_shuffle_product(list_1[1:], list_2, shuffle, shuffle_product)
            shuffle.pop()
            shuffle.append(list_2[0])
            _get_shuffle_product(list_1, list_2[1:], shuffle, shuffle_product)
            shuffle.pop()
max_prime_index = 25 # one and two digit primes.
max_element = 999
prime_set = set()
for p_index in range(1, max_prime_index+1):
    p = sympy.prime(p_index)
    for q_index in range(p_index, max_prime_index+1):
        q = sympy.prime(q_index)
        list_p = list(str(p))
        list_q = list(str(q))
        shuffle_product = get_shuffle_product(list_p, list_q)
        for s in shuffle_product:
            candidate = int(''.join(s))
            if sympy.isprime(candidate) and candidate <= max_element:
                prime_set.add(candidate)
print(sorted(prime_set))

from sympy import isprime
from itertools import chain, combinations
def powerset(s): # skipping empty set and entire set
    return chain.from_iterable(combinations(s, r) for r in range(1, len(s)))
def ok(n):
    if not isprime(n): return False
    s = str(n)
    for indices in powerset(range(len(s))):
        t1 = "".join(s[i] for i in indices)
        t2 = "".join(s[i] for i in range(len(s)) if i not in indices)
        if t1[0] != "0" and t2[0] != "0" and isprime(int(t1)) and isprime(int(t2)):
            return True
print([k for k in range(720) if ok(k)]) # Michael S. Branicky, Apr 16 2023

A083470 Smallest prime which is the concatenation of n distinct primes in increasing order.

Original entry on oeis.org

2, 23, 257, 2357, 235723, 23571143, 2357111371, 235711131767, 23571113172367, 2357111317192343, 235711131719233171, 23571113171923296797, 2357111317192329314189, 235711131719232931375979, 23571113171923293137414371, 2357111317192329313741436167

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), May 02 2003

Perhaps the n-th term begins with the concatenation of the first (n-1) primes, except for n = 3.

The 9th, 11th, 12th, 13th, .... terms are other counterexamples to the above "conjecture". On the other hand, it seems almost sure that the n-th term starts with the concatenation of the first (n-2) primes. - M. F. Hasler, Mar 20 2011

Cf. A069837, A083427.

More terms from David Wasserman, Nov 02 2004

A384726 a(n) is the least number that is both the product of n distinct primes and the concatenation of n distinct primes.

Original entry on oeis.org

2, 35, 273, 11235, 237615, 11237835, 1123317195, 111371237835, 11132343837615, 1113172923477615, 111317233377372295, 11131723677292413195, 1113172377671953734135, 111317192375336174123715

Offset: 1

Robert Israel, Jun 08 2025

a(n) is odd for n >= 2, because a number whose last digit is 2 and second-last is odd is divisible by 4.

			a(4) = 11235 is a term because 11235 is the product of four distinct primes 3, 5, 7, 107 and the concatenation of four distinct primes 11, 2, 3, 5, and no smaller number works.

Cf. A083427, A374665.

cdp:= proc(x, n, S)
  local i,y;
  if n = 1 then return (not(member(x,S)) and isprime(x)) fi;
  for i from 1 to ilog10(x)+2-n do
    y:= x mod 10^i;
    if member(y,S) or not isprime(y) then next fi;
    if procname((x-y)/10^i, n-1, S union {y}) then return true fi;
  od;
  false
end proc:
f:= proc(n) uses priqueue; local pq, t, p, x, i, L, v, Lp;
  initialize(pq);
  L:= [seq(ithprime(i), i=2..n+1)];
  v:= convert(L, `*`);
  insert([-v, L], pq);
  do
    t:= extract(pq);
    x:= -t[1];
    if cdp(x,n,{}) then return x fi;
    L:= t[2];
    p:= nextprime(L[-1]);
    for i from n to 1 by -1 do
      if i < n and L[i] <> prevprime(L[i+1]) then break fi;
      Lp:= [op(L[1..i-1]), op(L[i+1..n]), p];
      insert([-convert(Lp, `*`), Lp], pq)
  od od;
end proc:
f(1):= 2:
map(f, [$1..9]);

a(11)-a(14) from Jinyuan Wang, Jun 12 2025

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

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A361530 Primes that can be written as the result of shuffling the decimal digits of two primes.

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A083470 Smallest prime which is the concatenation of n distinct primes in increasing order.

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A384726 a(n) is the least number that is both the product of n distinct primes and the concatenation of n distinct primes.

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