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Base-3 deletable primes (written in base 10).
3

%I #54 Jan 14 2022 10:34:35

%S 2,5,7,11,17,19,23,29,47,53,59,61,71,73,83,89,101,107,137,167,173,179,

%T 181,191,197,223,233,251,263,269,317,431,461,491,503,509,521,541,547,

%U 557,569,587,593,653,659,673,677,683,701,709,719,809,911,947,953

%N Base-3 deletable primes (written in base 10).

%C A prime p is a base-b deletable prime if when written in base b it has the property that removing some digit leaves either the empty string or another deletable prime.

%C Deleting a digit cannot leave any leading zeros in the new string. For example, deleting the 2 in 2003 to obtain 003 is not allowed.

%H Robert Israel, <a href="/A319596/b319596.txt">Table of n, a(n) for n = 1..10000</a> (first 177 terms from Robert Price)

%p S:= {2}: count:= 0:

%p p:= 2;

%p while count < 200 do

%p p:= nextprime(p);

%p d:= floor(log[3](p));

%p for i from 0 to d do

%p x:= p mod 3^(i+1);

%p q:= (x mod 3^i) + (p-x)/3;

%p if q >= 3^(d-1) and member(q,S) then

%p S:= S union {p}; count:= count+1; break

%p fi

%p od;

%p od:

%p sort(convert(S,list)); # _Robert Israel_, Nov 26 2020

%t b = 3; d = {};

%t p = Select[Range[2, 10000], PrimeQ[#] &];

%t For[i = 1, i <= Length[p], i++,

%t c = IntegerDigits[p[[i]], b];

%t If[Length[c] == 1, AppendTo[d, p[[i]]]; Continue[]];

%t For[j = 1, j <= Length[c], j++,

%t t = Delete[c, j];

%t If[t[[1]] == 0, Continue[]];

%t If[MemberQ[d, FromDigits[t, b]], AppendTo[d, p[[i]]]; Break[]]]];

%t d (* _Robert Price_, Dec 05 2018 *)

%o (Python)

%o from sympy import isprime

%o from sympy.ntheory.digits import digits

%o def ok(n, base=3):

%o if not isprime(n): return False

%o if n < 3: return True

%o s = "".join(str(d) for d in digits(n, base)[1:])

%o si = (s[:i]+s[i+1:] for i in range(len(s)))

%o return any(t[0] != '0' and ok(int(t, base)) for t in si)

%o print([k for k in range(954) if ok(k)]) # _Michael S. Branicky_, Jan 14 2022

%Y Cf. A080608, A080603, A096235-A096246.

%K nonn,base,easy

%O 1,1

%A _Robert Price_, Nov 14 2018

%E Removed the term 3. As pointed out by _Kevin Ryde_, there is no need to "seed" the list using base-2 assumptions. - _Robert Price_, Dec 05 2018