A228323 -id:A228323 - cp's OEIS frontend

A228324 Primes arising from A228323 in order of their appearance.

13, 23, 29, 59, 521, 421, 47, 67, 613, 1013, 1019, 1619, 1627, 827, 811, 1511, 1523, 1223, 1217, 2017, 2029, 1429, 1433, 2633, 2647, 1847, 1831, 2531, 2539, 2239, 2237, 2437, 2441, 3041, 3049, 3449, 3457, 2857, 2843, 3643, 3659, 3259, 3251, 3851, 3853, 4253

Offset: 1

N. J. A. Sloane, Aug 20 2013

Eric Angelini, Posting to the Sequence Fans Mailing List, Aug 14 2013.

Michael S. Branicky, Table of n, a(n) for n = 1..10000
Index entries for primes involving decimal expansion of n

Cf. A228323.

from sympy import isprime
from itertools import islice
def c(s, t):
    u, v = int(s+t), int(t+s)
    if isprime(u): return u
    if isprime(v): return v
    return False
def agen():
    aset, k, mink = set(), 1, 2
    while True:
        an = k; aset.add(an); s, k = str(an), mink
        while k in aset or not c(s, str(k)): k += 1
        while mink in aset: mink += 1
        yield c(s, str(k))
print(list(islice(agen(), 46))) # Michael S. Branicky, Oct 17 2022

More terms from Alois P. Heinz, Aug 20 2013

A228325 a(n) is the smallest number m>n such that the concatenation nm is prime.

Original entry on oeis.org

3, 3, 7, 7, 9, 7, 9, 9, 11, 13, 17, 13, 19, 23, 23, 19, 21, 23, 31, 27, 29, 37, 33, 37, 31, 33, 29, 33, 39, 37, 37, 51, 43, 49, 39, 37, 39, 47, 43, 49, 53, 43, 49, 47, 47, 49, 51, 61, 51, 51, 53, 61, 81, 71, 57, 57, 79, 61, 81, 67, 63, 63, 67, 69, 69, 73, 79

Offset: 1

N. J. A. Sloane, Aug 20 2013

Max Alekseyev (see link in A068695) shows that a(n) always exists. - N. J. A. Sloane, Nov 13 2020

Suggested by the existence question in A228323.

			12 is not prime but 13 is, so a(1)=3.
23 is prime so a(2)=3.
34, 35, 36 are not prime but 37 is, so a(3)=7.

Cf. A228323, A068695.

smc[n_]:=Module[{m=n+1},If[OddQ[n],m++];While[!PrimeQ[n*10^IntegerLength[ m]+ m],m=m+2];m]; Array[smc,70] (* Harvey P. Dale, Apr 30 2016 *)

from sympy import isprime
from itertools import count
def a(n): return next(k for k in count(n+1) if isprime(int(str(n)+str(k))))
print([a(n) for n in range(1, 68)]) # Michael S. Branicky, Oct 18 2022

A338366 a(n) = smallest positive number k with all digits equal such that the concatenation k||n||k is prime, or -1 if no such k exists.

Original entry on oeis.org

1, 3, 7, 1, 11, 1, 7777, 3, 1, 1, 9, -1, 7, 33, 99, 1, 3, 1, 1, 9, 1, 11, -1, 1, 7, 3, 7777777777, 1111, 111, 1, 1, 3, 1, -1, 3, 33, 1, 3, 1, 77777777777777, 111, 3, 1111111111111111111111111111111111111111, 3, -1, 1, 3, 1, 1, 999, 7, 1, 11, 1, 7, -1, 33, 1, 3, 3, 1, 3, 1

Offset: 0

N. J. A. Sloane, Nov 08 2020

See A090287 for more information.
From Robert Price, Sep 20 2023: (Start)
For a(366), k is a string of 8441 1's.
The sequence then continues: 77, 1, 1, 3, 1, 1, 9, 7777777, 1, 11, 3, 1, 11, 9, 77, 11111, 1, 1, 33333, 3, 7, 9, 3, 1, 77, 1, 1, 9, 7777777777 until a(396) where k is a sequence of 269 1's.
The sequence then continues: 9, 777, 11, 9, 1, 7, 3, 7, 1, 11, 1, 1, 9, 9, 1111, 3, 999, 77777, 99, 7, 7, 3, 7, -1, 3, 1, 11, 77, 1, 77, 3, 1, 7, 3, 3, 1, 111111, 1, 7, 99, 7, 1111, 9, 1, 1, 11, 1, 7777777, 11, 1, 1111, 3, 1111, 7, 3, 7, 11, 3, 1, 1, 111, 3, 1, 3, 3, 1, 33, 9, 11, 33, 3, 7, 3, 3, 7, 99, 1, 1, 11, 3, 1, 9, 7, 77, 9, 1, 1, 3, 1, 7777, 33, 3, 1, 33, 3, 77, 77, 9, 1, 3, 33, 11111, 9, 9. (End)

			a(3) = 1 because 131 is prime.
a(4) = 11 because 11411 is prime, and all of 141, 242, 343, ..., 949 are composite.

Robert Price, Table of n, a(n) for n = 0..365
Index entries for primes involving decimal expansion of n

Cf. A090287.

Related sequences: A010785, A068695, A091088, A228323, A228325, A336893, A338712 (see also the Index link above).

More terms from Alois P. Heinz, Nov 08 2020

A246805 Lexicographically earliest sequence of distinct terms such that, when i

Original entry on oeis.org

1, 3, 4, 7, 19, 31, 67, 391, 583, 4549, 917467, 6777061, 86794921, 1421517037, 171234891469

Offset: 1

Paul Tek, Nov 16 2014

Two distinct terms can always be concatenated in some way to form a prime number.

Is this sequence infinite?

			The following concatenations are prime:
- j=2: a(1) U a(2)=13, a(2) U a(1)=31
- j=3: a(3) U a(1)=41, a(3) U a(2)=43
- j=4: a(1) U a(4)=17, a(4) U a(1)=71, a(2) U a(4)=37, a(4) U a(2)=73, a(3) U a(4)=47
- j=5: a(5) U a(1)=191, a(5) U a(2)=193, a(3) U a(5)=419, a(4) U a(5)=719, a(5) U a(4)=197
- j=6: a(1) U a(6)=131, a(6) U a(1)=311, a(2) U a(6)=331, a(6) U a(2)=313, a(3) U a(6)=431, a(6) U a(4)=317, a(5) U a(6)=1931, a(6) U a(5)=3119

Paul Tek, PARI program for this sequence

Cf. A156770, A228323.

PARI
```
See Link section.
```

from sympy import isprime
from itertools import islice
def c(s, slst):
    return all(isprime(int(s+t)) or isprime(int(t+s)) for t in slst)
def agen():
    slst, an, mink = [], 1, 2
    while True:
        yield an; slst.append(str(an)); an += 1
        while not c(str(an), slst): an += 1
print(list(islice(agen(), 10))) # Michael S. Branicky, Oct 17 2022

a(15) from Michael S. Branicky, Nov 07 2022

A306581 Lexicographically earliest sequence of distinct positive terms such that the binary representations of two consecutive terms can always been concatenated in some order, without leading zero, to produce the binary representation of a prime number.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 11, 8, 7, 9, 13, 10, 17, 12, 25, 18, 23, 15, 14, 19, 20, 21, 26, 27, 31, 29, 16, 37, 34, 45, 22, 39, 28, 55, 46, 57, 35, 24, 43, 36, 47, 33, 32, 41, 38, 67, 30, 53, 42, 61, 40, 49, 48, 73, 50, 51, 59, 56, 69, 44, 63, 52, 77, 60, 79, 54, 65

Offset: 1

Rémy Sigrist, Feb 25 2019

This sequence is the binary variant of A228323.

The sequence is well defined; the argument used to prove that A018800(n) always exists applies here also.

			The first terms, alongside their binary representations, and the concatenation of consecutive terms, with prime numbers denoted by a star, are:
  n   a(n)  bin(a(n))  bin(a(n)a(n+1))  bin(a(n+1)a(n))
  --  ----  ---------  ---------------  ---------------
   1     1          1              110              101*
   2     2         10             1011*            1110
   3     3         11            11100            10011*
   4     4        100           100101*          101100
   5     5        101           101110           110101*
   6     6        110          1101011*         1011110
   7    11       1011         10111000         10001011*
   8     8       1000          1000111*         1111000
   9     7        111          1111001          1001111*
  10     9       1001         10011101*        11011001

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of (n, a(n)) for n = 1..10000 (where the color corresponds to the parity of a(n): red for odd, blue for even)
Rémy Sigrist, Colored scatterplot of (n, a(n)-n) for n = 1..1000000 (where the color corresponds to the parity of a(n)-n: red for odd, blue for even)
Rémy Sigrist, Scatterplot of the first 1000000 terms of the ordinal transform of n -> a(n)-n
Rémy Sigrist, PARI program for A306581

See A228323 for the decimal variant.

Cf. A018800.

a = {1}; c[x_, y_] := FromDigits[Join @@ IntegerDigits[{x, y}, 2], 2]; While[Length@a < 67, j=1; While[MemberQ[a, j] || ! (PrimeQ@ c[a[[-1]], j] || PrimeQ@ c[j, a[[-1]]]), j++]; AppendTo[a, j]]; a (* Giovanni Resta, Feb 27 2019 *)

PARI
```
See Links section.
```

cp's OEIS Frontend

A228324 Primes arising from A228323 in order of their appearance.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Python

Extensions

A228325 a(n) is the smallest number m>n such that the concatenation nm is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A338366 a(n) = smallest positive number k with all digits equal such that the concatenation k||n||k is prime, or -1 if no such k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A246805 Lexicographically earliest sequence of distinct terms such that, when i

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Extensions

A306581 Lexicographically earliest sequence of distinct positive terms such that the binary representations of two consecutive terms can always been concatenated in some order, without leading zero, to produce the binary representation of a prime number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI