A329914 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.

21, 23, 27, 29, 33, 39, 57, 59, 69, 71, 83, 87, 99, 101, 107

Offset: 1

Author

Bernard Schott, Nov 24 2019

Keywords

• nonn
• base
• fini
• full

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

Examples

			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.
		

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)

Crossrefs

Cf. A000533, A329915 (corresponding numbers M).

Programs

Mathematica

Select[Range[21, 110], GCD[10, # - 10] == 1 && MemberQ[Mod[10^Range[#] + 1, # - 10], 0] &] (* Giovanni Resta, Nov 26 2019 *)