OFFSET
1,1
COMMENTS
The idea of this sequence comes from the 21-digit integer 112359550561797732809 in Penguin dictionary (see reference) with this property: "The smallest number which, when 1 is placed at both ends, the number is multiplied by 99". The terms of this sequence are the other numbers k that have the same property than 99 and the corresponding smallest numbers M in each set {M_k} are in A329915 (see link).
The Diophantine equation to solve is 1M1 = k * M with M that has q digits, this is equivalent to 10^(q+1) + 1 = (k-10) * M, with number of zeros in 10^(q+1) + 1 = q also.
Some results coming from this Diophantine equation:
q >= 2 and 21 <= k <= 110, so this sequence is finite. The integers (k-10) end with 1, 3, 7 or 9, hence k also.
Integer (k-10) must be a divisor of 10^(q+1)+1 = A000533(q+1).
For k = 21, there is 21 * 91 = 1[91]1 but also 21 * 9091 = 1[9091]1; hence, 91 and 9091 are terms of M_21.
Since 10^(q+1)+1 mod (k-10) is periodic and the period length cannot exceed k-10, it is easy to check that the sequence is indeed full. - Giovanni Resta, Nov 26 2019
REFERENCES
D. Wells, 112359550561797732809 entry, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1997, p. 196.
LINKS
Bernard Schott, Array with values of (q,k,M)
EXAMPLE
23 * 77 = 1[77]1, so k = 23 is a term and 13 * 77 = 1001; remark: number M = 77 has 2 digits and 10^3+1 has 2 zeros also.
29 * 52631579 = 1[52631579]1, so 29 is a term et 19 * 52631579 = 10^9 + 1 = 1000000001.
MATHEMATICA
Select[Range[21, 110], GCD[10, # - 10] == 1 && MemberQ[Mod[10^Range[#] + 1, # - 10], 0] &] (* Giovanni Resta, Nov 26 2019 *)
CROSSREFS
KEYWORD
nonn,base,fini,full
AUTHOR
Bernard Schott, Nov 24 2019
STATUS
approved