A116444 Numbers k which when sandwiched between two 9's give a multiple of k.

1, 3, 9, 11, 13, 21, 33, 39, 63, 77, 91, 99, 137, 219, 411, 657, 9091, 29703, 89109, 909091, 5882353, 10989011, 12145749, 12987013, 14354067, 20979021, 22556391, 32967033, 36437247, 38961039, 43062201, 52631579, 62937063, 67669173, 76923077, 90909091, 98901099

Offset: 1

Author

Giovanni Resta, Feb 15 2006

Keywords

• base
• nonn

Examples

			411 belongs since 94119 is a multiple of 411 (411*229).
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..5016

Crossrefs

Cf. A116436, A116437, A116438, A116439, A116440, A116441, A116442, A116443.

Programs

Mathematica

f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[9, 8] (* Ray Chandler, May 11 2007 *)
Select[Range[10^6],Mod[FromDigits[Join[{9},IntegerDigits[#],{9}]],#]==0&] (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Jan 20 2025 *)

PARI

for(d=1,10,fordiv(9*10^(d+1)+9,a,if(#Str(a)==d,print1(a", ")))) /* Martin Fuller, May 10 2007 */

Python

from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield from [1, 3, 9]
    for k in count(2):
        t = 9*(10**(k+1) + 1)
        yield from (t//i for i in range(900, 90, -1) if t%i == 0)
print(list(islice(agen(), 38))) # Michael S. Branicky, Mar 26 2023

Extensions

Corrected by Martin Fuller, May 10 2007

a(35) and beyond from Michael S. Branicky, Mar 26 2023