A116436 Numbers m which when sandwiched between two 1's give a multiple of m.

1, 11, 13, 77, 91, 137, 9091, 909091, 5882353, 10989011, 12987013, 52631579, 76923077, 90909091, 4347826087, 9090909091, 13698630137, 909090909091, 3448275862069, 10989010989011, 12987012987013, 76923076923077, 90909090909091, 9090909090909091, 909090909090909091

Offset

1,2

Comments

All k-digit numbers that divide 10^{k+1} + 1. - Franklin T. Adams-Watters, Apr 23 2008

Notice the infinite pattern m = (90..90..90)91 with 1m1/m = 21, e.g., 1911/91 = 190911/9091 = 19090911/909091 = 21 (see A095372). - Zak Seidov, Apr 22 2008

Corresponding numbers k such that k * a(n) = 1.a(n).1 where '.' stands for concatenation are in A351320. - Bernard Schott, Feb 07 2022

Links

Michael S. Branicky, Table of n, a(n) for n = 1..1441 (terms 1..87 from Franklin T. Adams-Watters)

Formula

A351320(n) * a(n) = 1.a(n).1 where "." stands for concatenation. - Bernard Schott, Feb 07 2022

Example

77 is a member since 1771 is a multiple of 77 (77*23).

Mathematica

f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[1, 14] (* Ray Chandler, May 11 2007 *)

Prog

(PARI) A116436(k) = {local(l, d, lb, ub); d=divisors(10^(k+1)+1); l=[]; lb=10^(k-1); ub=10*lb; for(i=1, #d, if(d[i]>=lb&&d[i]<ub, l=concat(l, [d[i]]))); l}
l=[]; for(i=1, 60, l=concat(l, A116436(i))); l
\\ Franklin T. Adams-Watters, Apr 22 2008
(Python)
from sympy import isprime
from itertools import count, islice
def agen(): # generator of terms
    yield 1
    for k in count(2):
        t = 10**(k+1) + 1
        yield from (t//i for i in range(100, 10, -1) if t%i == 0)
print(list(islice(agen(), 25))) # Michael S. Branicky, Mar 26 2023 following Franklin T. Adams-Watters but removing factorization

Crossrefs

Subsequence of A116437, A116438, A116439, A116440, A116441, A116442, A116443, A116444.

Cf. A136296, A329914, A329915, A351320.

Some subsequences, M such that k*M=1M1 for: A095372 \ {1} (k=21), A331630 (k=23), A351237 (k=83), A351238 (k=87), A351239 (k=101).

Sequence in context: A072580 A186640 A226242 * A185240 A056446 A023317

Adjacent sequences: A116433 A116434 A116435 * A116437 A116438 A116439

Keyword

base,nonn

Author

Giovanni Resta, Feb 15 2006

Status

approved