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

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, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 126, 130, 131, 140, 141, 150, 151

Offset

1,3

Comments

Inspired by the problem restricted to pandigital numbers suggested by E. Angelini (cf. link).

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

Links

Jayanta Basu, Table of n, a(n) for n = 1..1000

E. Angelini, 10 different digits, 9 products, seqfan list, Jan 03 2012.

Example

Any number having no two adjacent digits larger than 1 is trivially in the sequence.
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".

Maple

filter:= proc(n)
local L, S, i;
S:= convert(n, string);
for i from 1 to length(S)-1 do
  if StringTools:-Search(convert(parse(cat(S[i], "*", S[i+1])), string), S) = 0 then
      return false
  fi
od:
true
end proc:
select(filter, [$0..1000]); # Robert Israel, Oct 15 2014

Mathematica

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 *)

Prog

(PARI) has(n, m)={ my(p=10^#Str(m)); until( m>n\=10, n%p==m & return(1))}
is_A203565(n)={ my(d); for(i=2, #d=eval(Vec(Str(n))), has(n, d[i]*d[i-1]) | return); 1 }
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 */
for( n=0, 999, is_A203565(n) & print1(n", "))

Crossrefs

Cf. A203569 (digits are permutations of 0...n).

Cf. also A203566, A198298, A236402, A236403, A236404.

Cf. A227510 (product of all digits is a substring and > 0).

Sequence in context: A250395 A371123 A273880 * A132263 A089868 A089867

Adjacent sequences: A203562 A203563 A203564 * A203566 A203567 A203568

Keyword

nonn,base

Author

M. F. Hasler, Jan 03 2012

Status

approved