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A351320 a(n) is the unique integer k such that k * A116436(n) = 1.A116436(n).1 where "." stands for concatenation.

%I A351320 #19 Feb 10 2022 20:15:38
%S A351320 111,101,87,23,21,83,21,21,27,101,87,29,23,21,33,21,83,21,39,101,87,
%T A351320 23,21,21,21,83,101,87,59,23,21,99,57,21,27,21,101,87,29,23,21,83,69,
%U A351320 21,71,21,101,87,33,23,21,21,83,21,101,87,23,21,27,21,39,21,83,101,87,29,23,21,21,107,21,101
%N A351320 a(n) is the unique integer k such that k * A116436(n) = 1.A116436(n).1 where "." stands for concatenation.
%C A351320 Except for a(1) = 111, which is unique, all terms appear infinitely many times and belong to this set of fifteen integers: {21, 23, 27, 29, 33, 39, 57, 59, 69, 71, 83, 87, 99, 101, 107}; see A329914.
%C A351320 The corresponding indices where these integers appear the first time are respectively: 5, 4, 9, 12, 15, 19, 33, 29, 43, 45, 6, 3, 32, 2, 70.
%D A351320 David Wells, 112359550561797732809 entry, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1997, p. 196.
%F A351320 a(n) * A116436(n) = 1.A116436(n).1 where "." stands for concatenation.
%e A351320 A116436(1) = 1 and 111 * 1 = 1.1.1, hence a(1) = 111.
%e A351320 A116436(2) = 11 and 101 * 11 = 1.11.1, hence a(2) = 101.
%e A351320 A116436(32) = 112359550561797752809 and 99 * 112359550561797752809 = 1.112359550561797752809.1 hence a(32) = 99 (see Penguin reference).
%o A351320 (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}; \\ from A116436
%o A351320 a(n) = {my(v6=[], i=1); while (#v6 < n, v6 = concat(v6, A116436(i)); i++); my(x= v6[n]); my(k=1); while (eval(Str(1, x, 1)) % x, k++); eval(Str(1, x, 1))/x;} \\ _Michel Marcus_, Feb 10 2022
%Y A351320 Cf. A116436, A329914, A329915.
%Y A351320 M such that k*M=1M1 for: A095372 \ {1} (k=21), A331630 (k=23), A351237 (k=83), A351238 (k=87), A351239 (k=101).
%K A351320 nonn,base
%O A351320 1,1
%A A351320 _Bernard Schott_, Feb 07 2022