cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

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

Original entry on oeis.org

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

Views

Author

Lekraj Beedassy, Apr 11 2005

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Subsequence of A019549.
Cf. A121608, A121609, A121610, A083427, A129800.

Programs

  • Mathematica
    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 *)
  • Python
    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

Extensions

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

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

Views

Author

Eric Angelini, Oct 15 2009

Keywords

Comments

Leading zeros are not allowed (cf. A166504).
For any k > 0, there are A246806(k) terms with k digits. - Rémy Sigrist, Jan 08 2023

Examples

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

Links

Crossrefs

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

Programs

  • PARI
    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
  • Python
    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

Extensions

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

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

Views

Author

Colin Barker, Feb 17 2014

Keywords

Comments

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

Examples

			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.

Links

Crossrefs

Cf. A105184, A238057, A129800.
Cf. A010051, A000040.

Programs

  • Haskell
    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
  • Mathematica
    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 *)

A166503 Numbers k with property that k^2 is the concatenation of two or more prime numbers.

Original entry on oeis.org

5, 15, 17, 19, 23, 27, 49, 51, 53, 69, 73, 77, 85, 87, 107, 109, 115, 123, 141, 147, 151, 153, 157, 159, 163, 165, 171, 173, 177, 181, 187, 191, 199, 219, 229, 231, 233, 235, 239, 241, 243, 267, 269, 277, 279, 281, 289, 299, 319, 327, 331, 335, 337, 343, 357
Offset: 1

Views

Author

Zak Seidov, Oct 15 2009

Keywords

Comments

Only odd numbers are eligible.

Links

Crossrefs

Cf. A019549, A038692, A105184, A129800.

Programs

Formula

a(n) = sqrt(A038692(n)).

Extensions

Terms updated according to stricter definition of A152242 by M. F. Hasler, Oct 15 2009
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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