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A351238 Numbers M such that 87 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1.

%I A351238 #16 Feb 06 2022 23:08:40
%S A351238 13,12987013,12987012987013,12987012987012987013,
%T A351238 12987012987012987012987013,12987012987012987012987012987013,
%U A351238 12987012987012987012987012987012987013,12987012987012987012987012987012987012987013,12987012987012987012987012987012987012987012987013,12987012987012987012987012987012987012987012987012987013
%N A351238 Numbers M such that 87 * M = 1M1, where 1M1 denotes the concatenation of 1, M and 1.
%C A351238 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; 87 is the twelfth such integer, so 87 = A329914(12), and a(1) = A329915(12) = 13; hence, the terms of this sequence form the infinite set {M_87}.
%C A351238 Every term M = a(n) has q = 6*n-4 digits, and 10^(q+1)+1 that has q = 6*n-4 zeros in its decimal expansion is equal to 77 * M, so a(n) = M is a divisor of 10^(6*n-3)+1. Example: a(2) = 12987013 has 8 digits and 77 * 12987013 = 1000000001 that has 8 zeros in its decimal expansion.
%D A351238 D. Wells, 112359550561797732809 entry, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1997, p. 196.
%H A351238 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (1000001,-1000000).
%F A351238 a(n) = (10^(6*n-3)+1)/77 for n >= 1.
%e A351238 87 * 13 = 1[13]1, hence 13 is a term.
%e A351238 87 * 12987013 = 1[12987013]1, and 12987013 is a term.
%p A351238 seq((10^(6*n-3)+1)/77, n=1..15);
%t A351238 Table[(10^(6*n - 3) + 1)/77, {n, 1, 10}] (* _Amiram Eldar_, Feb 06 2022 *)
%Y A351238 Subsequence of A116436.
%Y A351238 Cf. A329914, A329915.
%Y A351238 Similar for: A095372 \ {1} (k=21), A331630 (k=23), A351237 (k=83), this sequence (k=87), A351239 (k=101).
%K A351238 nonn,base,easy
%O A351238 1,1
%A A351238 _Bernard Schott_, Feb 05 2022