A227510 Numbers such that product of digits of n is positive and a substring of n.

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, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 126, 131, 141, 151, 153, 161, 171, 181, 191, 211, 236, 243, 311, 315, 324, 362, 411, 511, 611, 612

Offset: 1

Author

Jayanta Basu, Jul 14 2013

Keywords

• nonn
• base

Comments

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.

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

You can also insert 1 in any position outside the substring that gives the product of digits. - Robert Israel, Aug 26 2014

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

Examples

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

Links

• Zak Seidov, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Maple

filter:= proc(n)
  local L;
  L:= convert(n,base,10);
  if has(L,0) then return false fi;
  verify(convert(convert(L,`*`),base,10),L,'sublist');
end proc:
select(filter, [$1..1000]); # Robert Israel, Aug 26 2014

Mathematica

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

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);}
n=1;while(n<10^4,if(isok(n),print1(n,", "));n++) \\ Derek Orr, Aug 26 2014

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

Python

from operator import mul
from functools import reduce
A227510 = [int(n) for n in (str(x) for x in range(1, 10**5)) if not n.count('0') and str(reduce(mul, (int(d) for d in n))) in n]
# Chai Wah Wu, Aug 26 2014

Extensions

Edited by M. F. Hasler, Oct 14 2014