A226108 -id:A226108 - cp's OEIS frontend

A378081 Primes that remain prime if any two of their digits are deleted.

223, 227, 233, 257, 277, 337, 353, 373, 523, 557, 577, 727, 733, 757, 773, 1117, 1171, 4111

Offset: 1

Enrique Navarrete, Nov 15 2024

Any term >= 1000 must have its last three digits be from {1, 3, 7, 9}. - Michael S. Branicky, Nov 15 2024
From David A. Corneth, Nov 16 2024: (Start)
Any term < 1000 has exactly three digits and all digits prime (cf. A019546).
Any term >= 1000 is of the form 100*p + r where p is prime and r has only digits coprime to 10 and 11 <= r <= 99.
Also digits in any term >= 1000 are from {1, 4, 7} and at most one digit from {2, 5, 8}. Else at least one of the numbers resulting from removing any two digits is a multiple of 3 and not 3 itself so not prime.
a(19) >= 10^9 if it exists. (End)
a(19) >= 10^10 if it exists. - Michael S. Branicky, Nov 16 2024
a(19) >= 10^100 if it exists. - David A. Corneth, Nov 18 2024

			From _David A. Corneth_, Nov 18 2024: (Start)
4111 is a term since 4111 is prime and removing any to digits from it gives 11 or 11 or 11 or 41 or 41 or 41 and all of those are prime.
No term can end in 12587 as by removing we can (among other numbers) obtain 287, 127, 128, 157, 158, 257 and 587 which are respectively 0 through 6 (mod 7) (all possible residue classes mod 7). So by prepending more digits to 12587 we can always get a multiple of 7 (that is larger than 7) after removing some two digits. (End)

David A. Corneth, PARI program

Cf. A019546, A051362, A226108.

q[n_] := Module[{d = IntegerDigits[n], nd}, nd = Length[d]; nd > 2 && AllTrue[FromDigits /@ Map[d[[#]] &, Subsets[Range[nd], {nd - 2}]], PrimeQ]]; Select[Prime[Range[600]], q] (* Amiram Eldar, Nov 16 2024 *)

PARI
```
\\ See Corneth link
```

from sympy import isprime
from itertools import combinations as C
def ok(n):
    if n < 100 or not isprime(n): return False
    s = str(n)
    return all(isprime(int(t)) for i, j in C(range(len(s)), 2) if (t:=s[:i]+s[i+1:j]+s[j+1:])!="")
print([k for k in range(1, 10**6) if ok(k)]) # Michael S. Branicky, Nov 15 2024

A378415 Primes with repeated digits that remain prime when any two of the same-valued digits are deleted.

Original entry on oeis.org

113, 131, 151, 211, 223, 227, 233, 277, 311, 337, 353, 373, 443, 557, 577, 599, 727, 733, 757, 773, 883, 887, 929, 997, 1009, 1013, 1021, 1031, 1051, 1103, 1117, 1123, 1129, 1153, 1171, 1213, 1223, 1229, 1231, 1291, 1373, 1399, 1447, 1471, 1531, 1553, 1559, 1663, 1667, 1669, 1733, 1777

Offset: 1

Enrique Navarrete, Nov 25 2024

Relaxed version of A378081, which contains only 18 terms up to 10^100.

Not a superset of A378081 since this sequence does not contain 257 and 523.

			114217 is in the sequence since deleting any two of the three 1's gives 4217 and 1427, both of which are prime.
131371 is not in the sequence since deleting the two 3's gives 1171, which is prime, but deleting two of the three 1's gives 3371, 3137, and 1337, the last one of which is not prime.

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Cf. A051362, A226108, A378081.

from sympy import isprime
from itertools import combinations as C
def ok(n):
    if n<100  or not isprime(n) or len(s:=str(n))==len(set(s)): return False
    return all(isprime(int(t)) for i, j in C(range(len(s)), 2) if s[i]==s[j] and (t:=s[:i]+s[i+1:j]+s[j+1:])!="")
print([k for k in range(1800) if ok(k)]) # Michael S. Branicky, Nov 25 2024

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A378081 Primes that remain prime if any two of their digits are deleted.

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