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

13, 12987013, 12987012987013, 12987012987012987013, 12987012987012987012987013, 12987012987012987012987012987013, 12987012987012987012987012987012987013, 12987012987012987012987012987012987012987013, 12987012987012987012987012987012987012987012987013, 12987012987012987012987012987012987012987012987012987013

Offset: 1

Author

Bernard Schott, Feb 05 2022

Keywords

• nonn
• base
• easy

Comments

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}.

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.

Examples

			87 * 13 = 1[13]1, hence 13 is a term.
87 * 12987013 = 1[12987013]1, and 12987013 is a term.
		

References

• D. Wells, 112359550561797732809 entry, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1997, p. 196.

Links

• Index entries for linear recurrences with constant coefficients, signature (1000001,-1000000).

Crossrefs

Subsequence of A116436.

Cf. A329914, A329915.

Similar for: A095372 \ {1} (k=21), A331630 (k=23), A351237 (k=83), this sequence (k=87), A351239 (k=101).

Programs

Maple

seq((10^(6*n-3)+1)/77, n=1..15);

Mathematica

Table[(10^(6*n - 3) + 1)/77, {n, 1, 10}] (* Amiram Eldar, Feb 06 2022 *)

Formula

a(n) = (10^(6*n-3)+1)/77 for n >= 1.