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A166504 Slime numbers: numbers which are the concatenation of primes, with "leading zeros" allowed.

%I A166504 #27 Jul 26 2024 22:11:05
%S A166504 2,3,5,7,11,13,17,19,22,23,25,27,29,31,32,33,35,37,41,43,47,52,53,55,
%T A166504 57,59,61,67,71,72,73,75,77,79,83,89,97,101,103,107,109,112,113,115,
%U A166504 117,127,131,132,133,135,137,139,149,151,157,163,167,172,173,175,177,179
%N A166504 Slime numbers: numbers which are the concatenation of primes, with "leading zeros" allowed.
%C A166504 A number is in this sequence if and only if it is prime or of the form a(k)*10^m+a(n), where a(k), a(n) are in this sequence and 10^m >= a(n) (and from this follows that one among a(k), a(n) can be taken to be prime).
%C A166504 This contains A152242 as a subsequence, but also additional terms like e.g. 202 which can be split into two primes, 2 and 02 (= 2). Such a splitting, where some of the substrings contain leading zeros, is not allowed in A152242.
%C A166504 Terms not in A152242 are listed in A166505.
%H A166504 Henri Picciotto, <a href="/A152242/a152242.pdf">Selected Integer Sequences</a>
%o A166504 (PARI) is_A166504(n)={ isprime(n) || ((bittest(n,0) || n%10==2) & for(i=1,#Str(n)-1, isprime(n%10^i) & is_A166504(n\10^i) & return(1)))}
%o A166504 (PARI) is(n)=if(isprime(n),return(1));if(n<202,return(isprime(n%10)&&isprime(n\10)));my(k=n%10,v);if(k==5||k==2,return(if(n<6,1,n\=10;has(n/10^valuation(n,10)))));if(k%2==0,return(0));v=digits(n);for(i=1,#v,if(isprime(n%10^i)&&is(n\10^i),return(1)));0 \\ _Charles R Greathouse IV_, Apr 30 2013
%Y A166504 Cf. A152242 (no leading zeros allowed).
%Y A166504 Cf. A085823 (super-slimes: all substrings are prime). - _Henri Picciotto_, Apr 01 2015
%K A166504 nonn,base
%O A166504 1,1
%A A166504 _M. F. Hasler_, Nov 02 2009
%E A166504 Edited by _Charles R Greathouse IV_, Apr 23 2010
%E A166504 Name "Slime numbers", after _Henri Picciotto_, added by _N. J. A. Sloane_, Mar 25 2015