A243162 Numbers n such that n^2 divides n.n.n where dot "." means concatenation.

1, 3, 13, 21, 37, 39, 91, 1443, 3367, 9901, 157737, 333667, 999001, 3075403, 9226209, 14287143, 33336667, 99990001, 1171182883, 1224848037, 1286294191, 1397863441, 1428557143, 1469179621, 1535254357, 1568996211, 1753536967, 1792076241, 1839599913, 1891910811

Offset: 1

Author

Hans Havermann, May 31 2014

Keywords

• nonn
• base

Comments

Number of d-digit solutions for d = 1..100: 2, 5, 0, 3, 0, 3, 2, 3, 0, 39, 0, 2, 0, 106, 0, 3, 3, 2, 0, 441, 4, 14, 0, 5, 0, 15, 2, 283, 0, 23, 0, 61, 0, 24, 21, 4, 0, 22, 0, 240, 0, 34, 0, 96, 3, 30, 0, 6, 16, 281, 0, 216, 0, 22, 5, 3894, 2, 10, 0, 149, 2, 11, 0, 407, 0, 25, 0, 2136, 0, 53983, 0, 12, 1, 29, 11, 1872, 99, 20, 0, 6984, 0, 45, 0, 279, 32, 10, 5, 15928, 0, 213, 24, 791, 0, 20, 14, 44, 0, 713, 12, 89804.

Numbers n such that n divides 100^d+10^d+1, where 10^(d-1)<=n<10^d. - Robert Israel, Jan 11 2017

Examples

			21^2 divides 212121; 91^2 divides 919191; so both 21 and 91 are in the sequence.
		

Links

• Hans Havermann, Table of n, a(n) for n = 1..1366

Crossrefs

Cf. A147553 (n^2 divides n.n), A147554 (primes in this sequence).

Contains A074992 and A168624.

Programs

Maple

Res:= {}:
for d from 1 to 15 do
  Res:= Res union select(t -> t >= 10^(d-1) and t < 10^d,
   numtheory:-divisors(100^d+10^d+1))
od:
sort(convert(Res,list)); # Robert Israel, Jan 11 2017

Mathematica

Do[d=Divisors[100^i+10^i+1];s=Select[d,Length[IntegerDigits[#]]==i&];If[Length[s]>0,Do[Print[s[[j]]],{j,Length[s]}]],{i,42}]