cp's OEIS Frontend

A203565 Numbers that contain the product of any two adjacent digits as a substring.

%I A203565 #24 Dec 23 2024 14:53:42
%S A203565 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,30,31,40,41,
%T A203565 50,51,60,61,70,71,80,81,90,91,100,101,102,103,104,105,106,107,108,
%U A203565 109,110,111,112,113,114,115,116,117,118,119,120,121,126,130,131,140,141,150,151
%N A203565 Numbers that contain the product of any two adjacent digits as a substring.
%C A203565 Inspired by the problem restricted to pandigital numbers suggested by E. Angelini (cf. link).
%C A203565 E. Angelini observes that up to a(86) this is the same as "Numbers that contain the product of (all) their digits as a substring" (cf. A227510 for the zeroless terms); then 212 is here but not there, and 236 is there and not here. - _M. F. Hasler_, Oct 14 2014
%H A203565 Jayanta Basu, <a href="/A203565/b203565.txt">Table of n, a(n) for n = 1..1000</a>
%H A203565 E. Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2012-January/016213.html">10 different digits, 9 products</a>, seqfan list, Jan 03 2012.
%e A203565 Any number having no two adjacent digits larger than 1 is trivially in the sequence.
%e A203565 The smallest nontrivial example is the number 126, which is in the sequence since 1*2=2 and 2*6=12 are both substrings of "126".
%p A203565 filter:= proc(n)
%p A203565 local L,S,i;
%p A203565 S:= convert(n,string);
%p A203565 for i from 1 to length(S)-1 do
%p A203565   if StringTools:-Search(convert(parse(cat(S[i],"*",S[i+1])),string),S) = 0 then
%p A203565       return false
%p A203565   fi
%p A203565 od:
%p A203565 true
%p A203565 end proc:
%p A203565 select(filter, [$0..1000]); # _Robert Israel_, Oct 15 2014
%t A203565 d[n_] := IntegerDigits[n]; Select[Range[0, 151], And @@ Table[MemberQ[FromDigits /@ Partition[d[#], IntegerLength[k], 1], k], {k, Times @@@ Partition[d[#], 2, 1]}] &] (* _Jayanta Basu_, Aug 10 2013 *)
%o A203565 (PARI) has(n,m)={ my(p=10^#Str(m)); until( m>n\=10, n%p==m & return(1))}
%o A203565 is_A203565(n)={ my(d); for(i=2,#d=eval(Vec(Str(n))), has(n,d[i]*d[i-1]) | return);1 }
%o A203565 is_A203565(n)={ my(d=Vecsmall(Str(n))); for(i=2,#d, d[i]<50 & i++ & next; has(n,d[i-1]%48*(d[i]-48)) | return);1 } /* twice as fast */
%o A203565 for( n=0,999, is_A203565(n) & print1(n","))
%Y A203565 Cf. A203569 (digits are permutations of 0...n).
%Y A203565 Cf. also A203566, A198298, A236402, A236403, A236404.
%Y A203565 Cf. A227510 (product of all digits is a substring and > 0).
%K A203565 nonn,base
%O A203565 1,3
%A A203565 _M. F. Hasler_, Jan 03 2012