A096246 - OEIS

%I #24 Jan 13 2022 18:40:00

%S 2,3,5,7,11,13,19,23,29,37,43,47,53,59,61,73,79,83,101,107,109,137,

%T 149,151,157,163,167,173,179,197,211,229,277,281,293,307,311,313,317,

%U 331,347,349,359,389,397,419,421,457,461,467,557,563,569,587,599,601,613

%N Base-2 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. However, in base 2 we adopt the convention that 2 = 10 and 3 = 11 are deletable.

%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 Lei Zhou, <a href="/A096246/b096246.txt">Table of n, a(n) for n = 1..10000</a>

%p isDel := proc(n::integer) local b2,redu,rpr,d; if n = 2 or n =3 then RETURN(true); elif not isprime(n) then RETURN(false); else b2 := convert(n,base,2); for d from 1 to nops(b2) do redu := [op(1..d-1,b2),op(d+1..nops(b2),b2) ]; if op(nops(redu),redu) = 1 then rpr := sum( op(i,redu)*2^(i-1),i=1..nops(redu)); if isDel(rpr) then RETURN(true); fi; fi; od; RETURN(false); fi; end: for n from 1 to 200 do if isDel(ithprime(n)) then printf("%d,",ithprime(n)); fi; od: # _R. J. Mathar_, Apr 25 2006

%t a = {}; c = {1}; While[Length[a] < 100, b = c; c = {}; lb = Length[b]; Do[nb = b[[ib]]; cdb = RealDigits[nb, 2]; db = cdb[[1]]; ldb = cdb[[2]]; Do[dc = Insert[db, 0, j]; nc = FromDigits[dc, 2]; If[PrimeQ[nc], AppendTo[c, nc]], {j, 2, ldb + 1}]; Do[dc = Insert[db, 1, j]; nc = FromDigits[dc, 2]; If[PrimeQ[nc], AppendTo[c, nc]], {j, 2, ldb + 1}], {ib, 1, lb}]; c = Union[{}, c]; a = Union[a, c]]; a (* _Lei Zhou_, Mar 06 2015 *)

%t a = {0, 2}; d = {2, 3};

%t For[n = 3, n <= 15, n++,

%t   p = Select[Range[2^(n - 1), 2^n - 1], PrimeQ[#] &];

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

%t    c = IntegerDigits[p[[i]], 2];

%t    For[j = 1, j <= n, j++,

%t     t = Delete[c, j];

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

%t     If[MemberQ[d, FromDigits[t, 2]], AppendTo[d, p[[i]]];  Break[]]]]];

%t d (* _Robert Price_, Nov 11 2018 *)

%o (Python)

%o from sympy import isprime

%o def ok(n):

%o     if not isprime(n): return False

%o     if n == 2 or n == 3: return True

%o     b = bin(n)[2:]

%o     bi = (b[:i]+b[i+1:] for i in range(len(b)))

%o     return any(t[0] != '0' and ok(int(t, 2)) for t in bi)

%o print([k for k in range(614) if ok(k)]) # _Michael S. Branicky_, Jan 13 2022

%Y Cf. A080608, A080603, A096235-A096245.

%K nonn,base,easy

%O 1,1

%A _Michael Kleber_, Feb 28 2003

%E More terms from _R. J. Mathar_, Apr 25 2006