A038692 -id:A038692 - cp's OEIS frontend

A152242 Integers formed by concatenating primes.

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

A038840 Cubes that are concatenations of primes.

Original entry on oeis.org

27, 343, 729, 1331, 2197, 3375, 4913, 5832, 19683, 21952, 24389, 29791, 35937, 54872, 59319, 91125, 103823, 110592, 117649, 148877, 166375, 195112, 205379, 226981, 250047, 314432, 357911, 389017, 421875, 571787, 614125, 681472, 753571, 857375, 941192, 1092727

Offset: 1

Jeff Burch

			Examples from _Sean A. Irvine_, Feb 15 2021: (Start)
3^3 = 27 = (2)(7).
18^3 = 5832 = (5)(83)(2).
29^3 = 24389 = (2)(43)(89).
(End)

Sean A. Irvine, Java program (github)

Cf. A038692.

for(n=2,1e2,if(is_A152242(n^3),print1(n^3", "))) \\ Charles R Greathouse IV, Jun 10 2013

Missing a(6), a(7), a(9), a(13), a(16)-a(18), a(21), a(23) from Charles R Greathouse IV, Jun 10 2013

Missing terms inserted and more terms from Sean A. Irvine, Feb 15 2021

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

Zak Seidov, Oct 15 2009

Only odd numbers are eligible.

Jinyuan Wang, Table of n, a(n) for n = 1..10000

Cf. A019549, A038692, A105184, A129800.

for(n=1, 9999, is_A152242(n^2) & print1(n, ", ")) \\ M. F. Hasler, Oct 15 2009

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

Terms updated according to stricter definition of A152242 by M. F. Hasler, Oct 15 2009

A351925 Squares which are the concatenation of two primes.

Original entry on oeis.org

25, 289, 361, 529, 729, 2401, 2601, 2809, 4761, 5329, 5929, 7569, 11449, 11881, 15129, 19881, 21609, 22801, 23409, 24649, 25281, 26569, 29241, 29929, 31329, 34969, 36481, 39601, 47961, 52441, 53361, 54289, 57121, 58081, 59049, 71289, 77841, 83521, 89401

Offset: 1

Max Z. Scialabba, Feb 25 2022

The first term that is the concatenation of two primes in more than one way is a(11) = 5929 = 5 | 929 = 59 | 29. - Robert Israel, Oct 01 2023

			25 is the concatenation of 2 and 5, both primes.
4761 is the concatenation of 47 and 61, both primes.

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

Cf. A000290 (squares), A039686, A106582, inverse of A167535.

Cf. A038692, A225135.

L:= NULL: count:=0:
for x from 1 by 2 while count < 100 do
  xs:= x^2;
  for i from 1 to ilog10(xs) do
    a:= xs mod 10^i;
    if a > 10^(i-1) and isprime(a) then
      b:= (xs-a)/10^i;
      if isprime(b) then
        L:= L, xs; count:= count+1; break
      fi fi
od od:
L; # Robert Israel, Oct 01 2023

isb(n)={my(d=10); while(dAndrew Howroyd, Feb 26 2022

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    for k in count(1):
        s = str(k*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*k
print(list(islice(agen(), 39))) # Michael S. Branicky, Feb 26 2022

Intersection of A106582 and A000290.

A165631 Numbers whose cube is a concatenation of primes, i.e., in A152242.

Original entry on oeis.org

3, 7, 9, 11, 13, 15, 17, 18, 27, 28, 29, 31, 33, 38, 39, 45, 47, 48, 49, 53, 55, 58, 59, 61, 63, 68, 71, 73, 75, 83, 85, 88, 91, 95, 98, 103, 108, 111, 113, 117, 121, 125, 127, 131, 133, 135, 137, 138, 148, 153, 157, 159, 161, 163, 167, 168, 173, 175, 177, 178, 179

Offset: 1

Zak Seidov and M. F. Hasler, Oct 16 2009

Cf. A038692, A019549, A105184, A166503.

for(n=1,999, is_A152242(n^3) & print1(n", "))

Edited by Charles R Greathouse IV, Apr 24 2010

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A152242 Integers formed by concatenating primes.

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A038840 Cubes that are concatenations of primes.

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

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A351925 Squares which are the concatenation of two primes.

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A165631 Numbers whose cube is a concatenation of primes, i.e., in A152242.

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