A166507 Least n-comma number: smallest nonnegative integer that occurs in the sequence S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]) for at least n different splittings a(n)=concat(S[0],S[1]).

0, 10, 1023, 2676, 16867, 111688, 1522828, 11386882, 112273999, 1212143716, 11232152998, 121321194596

Offset: 0

Author

Eric Angelini and M. F. Hasler, Oct 29 2009

Keywords

• base
• hard
• more
• nonn

Comments

This subsequence of A166508 and of A166511 consists of the least numbers (= nonnegative integers) a(n) which occur as term in the sequence S(a,b), defined by S[0]=a, S[1]=b, S[k+1]=S[k]+10*last_digit(S[k-1])+first_digit(S[k]), for at least n legal splittings a(n)=concat(a,b).

"Legal" means that a and b have at least one digit each, and b has no leading zero(s) (unless b=0). Therefore a(n) must have at least n nonzero digits preceding the last digit (cf. formula). See A166511 and A166512 for more information.

Examples

			There are 0 ways to split a(0)=0 in two substrings, so this is the smallest 0-comma number.
The number a(1)=10 is the smallest 1-comma number, cf. A166511.
The number a(2)=1023 is the smallest 2-comma number: it occurs in S(10,23) and in S(102,3), cf. A166512.
		

Links

• E. Angelini, k-comma numbers, Oct. 2009.
• E. Angelini, k-comma numbers [Cached copy, with permission]

Crossrefs

Cf. A166508, A166511, A166512, A166513.

Programs

PARI

A166507(k) = { my(a,b,c); for( n=10^k\9*10,1e9, c=k; n%100 | next; for(d=1,#Str(n)-1, d+c>#Str(n) & break /* not possible: next n */; a=n\10^d, b=n%10^d; b<10^(d-1) & d>1 & next /* not legal: next d */; while(n > b=10*(a%10)+b\10^(#Str(b)-1)+a=b,); b>n & next; c-- | return(n)))}

Formula

a(k) >= [10^k/9]*10 = (10^(k+1)-1)/9-1.

Extensions

a(6)-a(11) from Lars Blomberg, Jan 06 2015