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

%I A116436 #38 Mar 26 2023 10:04:46
%S A116436 1,11,13,77,91,137,9091,909091,5882353,10989011,12987013,52631579,
%T A116436 76923077,90909091,4347826087,9090909091,13698630137,909090909091,
%U A116436 3448275862069,10989010989011,12987012987013,76923076923077,90909090909091,9090909090909091,909090909090909091
%N A116436 Numbers m which when sandwiched between two 1's give a multiple of m.
%C A116436 All k-digit numbers that divide 10^{k+1} + 1. - _Franklin T. Adams-Watters_, Apr 23 2008
%C A116436 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
%C A116436 Corresponding numbers k such that k * a(n) = 1.a(n).1 where '.' stands for concatenation are in A351320. - _Bernard Schott_, Feb 07 2022
%H A116436 Michael S. Branicky, <a href="/A116436/b116436.txt">Table of n, a(n) for n = 1..1441</a> (terms 1..87 from Franklin T. Adams-Watters)
%F A116436 A351320(n) * a(n) = 1.a(n).1 where "." stands for concatenation. - _Bernard Schott_, Feb 07 2022
%e A116436 77 is a member since 1771 is a multiple of 77 (77*23).
%t A116436 f[k_, d_] := Flatten@Table[Select[Divisors[k*(10^(i + 1) + 1)], IntegerLength[ # ] == i &], {i, d}]; f[1, 14] (* _Ray Chandler_, May 11 2007 *)
%o A116436 (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}
%o A116436 l=[];for(i=1,60,l=concat(l,A116436(i))); l
%o A116436 \\ _Franklin T. Adams-Watters_, Apr 22 2008
%o A116436 (Python)
%o A116436 from sympy import isprime
%o A116436 from itertools import count, islice
%o A116436 def agen(): # generator of terms
%o A116436     yield 1
%o A116436     for k in count(2):
%o A116436         t = 10**(k+1) + 1
%o A116436         yield from (t//i for i in range(100, 10, -1) if t%i == 0)
%o A116436 print(list(islice(agen(), 25))) # _Michael S. Branicky_, Mar 26 2023 following Franklin T. Adams-Watters but removing factorization
%Y A116436 Subsequence of A116437, A116438, A116439, A116440, A116441, A116442, A116443, A116444.
%Y A116436 Cf. A136296, A329914, A329915, A351320.
%Y A116436 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).
%K A116436 base,nonn
%O A116436 1,2
%A A116436 _Giovanni Resta_, Feb 15 2006