A121608 -id:A121608 - 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

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.

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.

A342049 Primes formed by the concatenation of exactly two consecutive composite numbers.

Original entry on oeis.org

89, 5051, 5657, 6263, 6869, 8081, 9091, 9293, 120121, 186187, 188189, 200201, 216217, 242243, 246247, 252253, 278279, 300301, 308309, 318319, 338339, 342343, 350351, 362363, 368369, 390391, 402403, 410411, 416417, 426427, 428429, 440441, 446447, 450451, 452453, 470471, 476477, 482483

Offset: 1

Bernard Schott, Feb 26 2021

When a prime is obtained by the concatenation of exactly two consecutive composite numbers, the first one always ends with 0, 2, 6, 8 while the second one ends respectively with 1, 3, 7, 9.
a(1) = 89 is also the smallest prime whose digits are composite (A051416).
a(n) has an even number of digits. If it would have an odd number of digits then it is like 99..99100..00 but that is composite. - David A. Corneth, Feb 27 2021

			If (2,q) is the smallest term formed by the concatenation of 2 consecutive composite numbers with each q digits: (2,1) = a(1) = 89, (2,2) = a(2) = 5051, (2,3) = a(9) = 120121, (2,4) = 10021003, (2,5) = 1001010011, (2,6) = 100010100011.

G. L. Honaker, Jr. and Chris Caldwell, Prime Curios! 89.

Cf. A002808, A087341.

Subsequence of A030458 and A121608.

isc(c) = (c>1) && ! isprime(c);
isok(p) = {if (isprime(p), my(d=digits(p)); for (i=1, #d-1, my(b = fromdigits(vector(i, k, d[k]))); if (d[i+1], my(c = fromdigits(vector(#d-i, k, d[k+i]))); if (isc(b) && isc(c) && ((primepi(c) - primepi(b)) == c-b-1), return (1)); ); ); ); } \\ Michel Marcus, Feb 27 2021

first(n) = { pc = 4; my(res = vector(n)); t = 0; forcomposite(c = 6, oo, nc = pc * 10^#digits(c) + c; if(isprime(nc), t++; res[t] = nc; if(t >= n, return(res) ) ); pc = c; ) } \\ David A. Corneth, Feb 27 2021

is(n) = { my(d = digits(n)); if(#d % 2 == 1, return(0) ); fc = fromdigits(vector(#d \ 2, i, d[i])); lc = fromdigits(vector(#d \ 2, i, d[i+#d\2])); lc - fc == 1 && !isprime(fc) && !isprime(lc) && nextprime(fc)==nextprime(lc) && isprime(n) } \\ David A. Corneth, Feb 27 2021

from sympy import isprime
def agento(lim):
  digs, pow10 = 1, 10
  while True:
    for c2 in range(max(pow10//10+1, 3), pow10, 2):
      if not isprime(c2) and not isprime(c2-1):
        c1c2 = (c2-1)*pow10+c2
        if c1c2 > lim: return
        if isprime(c1c2): yield c1c2
    digs, pow10 = digs+1, pow10*10
print([an for an in agento(482483)]) # Michael S. Branicky, Feb 27 2021

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

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A121609 Composite numbers that can be written as concatenation of two primes 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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A342049 Primes formed by the concatenation of exactly two consecutive composite numbers.

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