%I #29 Oct 15 2014 02:55:39

%S 1,2,3,4,5,6,7,8,9,11,12,13,14,15,16,17,18,19,21,31,41,51,61,71,81,91,

%T 111,112,113,114,115,116,117,118,119,121,126,131,141,151,153,161,171,

%U 181,191,211,236,243,311,315,324,362,411,511,611,612

%N Numbers such that product of digits of n is positive and a substring of n.

%C All numbers with at least one zero digit have a product of digits which is a substring; these have been kept out by the restriction on positivity.

%C The sequence is infinite: if n is a term 10n+1 is also a term. Are there any other patterns (except for prepending 1 to any term)? - _Zak Seidov_, Jul 24 2013

%C You can also insert 1 in any position outside the substring that gives the product of digits. - _Robert Israel_, Aug 26 2014

%C See also A203566 for a nontrivial subsequence of A203565. The zeroless members of the latter differ from this sequence from 212 on which is there but not here, while 236 is the first here but not there. - _M. F. Hasler_, Oct 14 2014

%H Zak Seidov, <a href="/A227510/b227510.txt">Table of n, a(n) for n = 1..10000</a>

%e The product of the digits of 236 is 36, a substring of 236, and hence 236 is a member.

%p filter:= proc(n)

%p local L;

%p L:= convert(n,base,10);

%p if has(L,0) then return false fi;

%p verify(convert(convert(L,`*`),base,10),L,'sublist');

%p end proc:

%p select(filter, [$1..1000]); # _Robert Israel_, Aug 26 2014

%t Select[Range[650], FreeQ[x = IntegerDigits[#], 0] && MemberQ[FromDigits /@ Partition[x, IntegerLength[y = Times @@ x], 1], y] &]

%o (Python)

%o from operator import mul

%o from functools import reduce

%o A227510 = [int(n) for n in (str(x) for x in range(1,10**5)) if

%o ..........not n.count('0') and str(reduce(mul,(int(d) for d in n))) in n]

%o # _Chai Wah Wu_, Aug 26 2014

%o (PARI) {isok(n)=d=digits(n);p=prod(i=1,#d,d[i]);k=1;while(p&&k<=(#d-#digits(p)+1),v=[];for(j=k,k+#digits(p)-1,v=concat(v,d[j]));if(v==digits(p),return(1));k++);return(0);}

%o n=1;while(n<10^4,if(isok(n),print1(n,", "));n++) \\ _Derek Orr_, Aug 26 2014

%o (PARI) is_A227510(n)={(t=digits(prod(i=1,#n=digits(n),n[i])))&&for(i=0,#n-#t,vecextract(n,2^(i+#t)-2^i)==t&&return(1))} \\ _M. F. Hasler_, Oct 14 2014

%Y Cf. A052018, A203565, A203566, A203569, A198298, A236402, A236403, A236404.

%K nonn,base

%O 1,2

%A _Jayanta Basu_, Jul 14 2013

%E Edited by _M. F. Hasler_, Oct 14 2014