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

%I A378081 #32 Nov 20 2024 09:59:40
%S A378081 223,227,233,257,277,337,353,373,523,557,577,727,733,757,773,1117,
%T A378081 1171,4111
%N A378081 Primes that remain prime if any two of their digits are deleted.
%C A378081 Any term >= 1000 must have its last three digits be from {1, 3, 7, 9}. - _Michael S. Branicky_, Nov 15 2024
%C A378081 From _David A. Corneth_, Nov 16 2024: (Start)
%C A378081 Any term < 1000 has exactly three digits and all digits prime (cf. A019546).
%C A378081 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.
%C A378081 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.
%C A378081 a(19) >= 10^9 if it exists. (End)
%C A378081 a(19) >= 10^10 if it exists. - _Michael S. Branicky_, Nov 16 2024
%C A378081 a(19) >= 10^100 if it exists. - _David A. Corneth_, Nov 18 2024
%H A378081 David A. Corneth, <a href="/A378081/a378081.gp.txt">PARI program</a>
%e A378081 From _David A. Corneth_, Nov 18 2024: (Start)
%e A378081 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.
%e A378081 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)
%t A378081 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 *)
%o A378081 (Python)
%o A378081 from sympy import isprime
%o A378081 from itertools import combinations as C
%o A378081 def ok(n):
%o A378081     if n < 100 or not isprime(n): return False
%o A378081     s = str(n)
%o A378081     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:])!="")
%o A378081 print([k for k in range(1, 10**6) if ok(k)]) # _Michael S. Branicky_, Nov 15 2024
%o A378081 (PARI) \\ See Corneth link
%Y A378081 Cf. A019546, A051362, A226108.
%K A378081 nonn,base,more
%O A378081 1,1
%A A378081 _Enrique Navarrete_, Nov 15 2024