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A080608
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Deletable primes: primes such that removing some digit leaves either the empty string or another deletable prime (possibly preceded by some zeros).
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44
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2, 3, 5, 7, 13, 17, 23, 29, 31, 37, 43, 47, 53, 59, 67, 71, 73, 79, 83, 97, 103, 107, 113, 127, 131, 137, 139, 157, 163, 167, 173, 179, 193, 197, 223, 229, 233, 239, 263, 269, 271, 283, 293, 307, 311, 313, 317, 331, 337, 347, 353, 359, 367, 373, 379, 383, 397, 431, 433, 439
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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Subsequence of A179336. - Reinhard Zumkeller, Jul 11 2010
Leading zeros are allowed in the number that appears after the digit is deleted. For example the prime 100003 is deletable because of the sequence 00003, 0003, 003, 03, 3 consists of primes. Because of this, it appears that deletable primes are relatively common in the region just above a power of ten. For example 10^1000 + 2713 is a deletable prime. - Jeppe Stig Nielsen, Aug 01 2018
For a version that does not allow leading zeros, see A305352. - Jeppe Stig Nielsen, Aug 01 2018
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LINKS
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David W. Wilson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Deletable Prime
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EXAMPLE
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410256793 is a deletable prime since each member of the sequence 410256793, 41256793, 4125673, 415673, 45673, 4567, 467, 67, 7 is prime (Weisstein, Caldwell).
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MAPLE
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read("transforms"):
isA080608 := proc(n)
option remember;
local dgs, i ;
if isprime(n) then
if n < 10 then
true;
else
dgs := convert(n, base, 10) ;
for i from 1 to nops(dgs) do
subsop(i=NULL, dgs) ;
digcatL(ListTools[Reverse](%)) ;
if procname(%) then
return true;
end if;
end do:
false ;
end if;
else
false;
end if;
end proc:
n := 1;
for i from 1 to 500 do
p := ithprime(i) ;
if isA080608(p) then
printf("%d %d\n", n, p) ;
n := n+1 ;
fi ;
end do: # R. J. Mathar, Oct 11 2014
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MATHEMATICA
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Rest@ Union@ Nest[Function[{a, p}, Append[a, With[{w = IntegerDigits[p]}, If[# == True, p, 0] &@ AnyTrue[Array[FromDigits@ Delete[w, #] &, Length@ w], ! FreeQ[a, #] &]]]] @@ {#, Prime[Length@ # + 1]} &, Prime@ Range@ PrimePi@ 10, 81] (* Michael De Vlieger, Aug 02 2018 *)
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PROG
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(PARI) is(n) = !ispseudoprime(n)&&return(0); my(d=digits(n)); #d==1&&return(1); for(i=1, #d, is(fromdigits(vecextract(d, Str("^"i))))&&return(1)); 0 \\ Jeppe Stig Nielsen, Aug 01 2018
(Perl) use ntheory ":all"; sub is { my $n=shift; return 0 unless is_prime($n); my @d=todigits($n); return 1 if @d==1; is(fromdigits([vecextract(\@d, ~(1<<$_))])) && return 1 for 0..$#d; 0; } # Dana Jacobsen, Nov 16 2018
(Python)
from sympy import isprime, prevprime
def ok(n):
if not isprime(n): return False
if n < 10: return True
s = str(n)
si = (s[:i]+s[i+1:] for i in range(len(s)))
return any(t[0] != '0' and ok(int(t)) for t in si)
print([k for k in range(440) if ok(k)]) # Michael S. Branicky, Jan 13 2022
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CROSSREFS
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Cf. A080603, A096235-A096246, A305352.
Sequence in context: A247980 A234851 A179336 * A305352 A347864 A137812
Adjacent sequences: A080605 A080606 A080607 * A080609 A080610 A080611
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KEYWORD
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nonn,easy,base
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AUTHOR
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David W. Wilson, Feb 25 2003
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STATUS
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approved
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