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A351237 Numbers M such that 83 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1.

%I A351237 #22 Nov 01 2022 19:38:25
%S A351237 137,13698630137,1369863013698630137,136986301369863013698630137,
%T A351237 13698630136986301369863013698630137,
%U A351237 1369863013698630136986301369863013698630137,136986301369863013698630136986301369863013698630137
%N A351237 Numbers M such that 83 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1.
%C A351237 There are only 15 numbers k such that there exist numbers M_k which, when 1 is placed at both ends of M_k, the number M_k is multiplied by k; 83 is the eleventh such integer, so 83 = A329914(11), and a(1) = A329915(11) = 137; hence, the terms of this sequence form the infinite set {M_83}.
%C A351237 Every term M = a(n) has q = 8*n-5 digits, and 10^(q+1)+1 that has q = 8*n-5 zeros in its decimal expansion is equal to 73 * M, so a(n) = M is a divisor of 10^(8*n-4)+1. Example: a(2) = 13698630137 has 11 digits and 73 * 13698630137 = 1000000000001 that has 11 zeros in its decimal expansion.
%D A351237 D. Wells, 112359550561797732809 entry, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1997, p. 196.
%H A351237 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (100000001,-100000000).
%F A351237 a(n) = (10^(8*n-4)+1)/73 for n >= 1.
%e A351237 83 * 137 = 1[137]1, hence 137 is a term.
%e A351237 83 * 13698630137 = 1[13698630137]1, and 13698630137 is another term.
%p A351237 seq((10^(8*n-4)+1)/73, n=1..15);
%t A351237 Table[(10^(8*n-4)+1)/73, {n, 1, 7}] (* _Amiram Eldar_, Feb 06 2022 *)
%t A351237 LinearRecurrence[{100000001,-100000000},{137,13698630137},20] (* _Harvey P. Dale_, Nov 01 2022 *)
%Y A351237 Subsequence of A116436.
%Y A351237 Cf. A329914, A329915.
%Y A351237 Similar for: A095372 \ {1} (k=21), A331630 (k=23), this sequence (k=83), A351238 (k=87), A351239 (k=101).
%K A351237 nonn,base,easy
%O A351237 1,1
%A A351237 _Bernard Schott_, Feb 05 2022