A169819 a(n) = total number of distinct divisors of n and all of its substrings.

1, 2, 2, 3, 2, 4, 2, 4, 3, 5, 2, 6, 3, 5, 4, 7, 3, 8, 4, 7, 5, 4, 4, 8, 4, 6, 6, 7, 5, 9, 3, 7, 4, 6, 5, 9, 4, 7, 5, 9, 4, 9, 5, 6, 8, 7, 5, 10, 7, 7, 5, 7, 4, 10, 4, 11, 6, 7, 5, 13, 5, 6, 8, 9, 7, 8, 6, 9, 7, 9, 3, 13, 4, 6, 7, 9, 4, 11, 5, 11, 8, 6, 6, 13, 7, 8, 8, 8, 7, 13, 6, 8, 5, 7, 6, 13, 5, 10, 6

Offset: 1

Author

Zak Seidov, May 28 2010

Keywords

• base
• nonn

Comments

Note that we are counting 0 when it occurs as a digit of n, but are not counting any other integers as divisors of 0. (If we did, there would be infinitely many of them; every integer divides 0.) [From Franklin T. Adams-Watters, May 29 2010]

Examples

			a(56) = 11 because divisors of 56 are d1= {1, 2, 4, 7, 8, 14, 28, 56}; 56 has two substrings 5,6; divisors of 5 are d2= {1, 5}, and divisors of 6 are d3= {1, 2, 3,6} ; union of d1,d2,d3 gives 11 distinct divisors of 56 and all of its substrings: {1, 2, 3, 4, 5, 6, 7, 8, 14, 28, 56}.
		

Crossrefs

Cf. A177834

Programs

Mathematica

Table[id = IntegerDigits[n]; FLA = Flatten[Table[Partition[id, k, 1], {k, Length[id]}], 1]; fd = Union[FromDigits /@ FLA]; dv = Length[Union[Flatten[Divisors /@ fd]]], {n, 200}]