A306355 Numbers k such that the period of 1/k, or 0 if 1/k terminates, is strictly greater than the period of the decimal expansion of 1/m for all m < k.

1, 3, 7, 17, 19, 23, 29, 47, 59, 61, 97, 109, 113, 131, 149, 167, 179, 181, 193, 223, 229, 233, 257, 263, 269, 289, 313, 337, 361, 367, 379, 383, 389, 419, 433, 461, 487, 491, 499, 503, 509, 541, 571, 577, 593, 619, 647, 659, 701, 709, 727, 743, 811, 821, 823

Offset: 1

Author

Matthew Schulz, Feb 09 2019

Keywords

• nonn
• base

Comments

This sequence is infinite because 1/(10^k-1) has a period of k for all k, so the period can be arbitrarily large.

Are 1, 3, 289 and 361 the only terms that are not in A001913? - Robert Israel, Feb 10 2019

Examples

			7 is a term because 1/7 has a period of 6, which is greater than the periods of 1/m for m < 7.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Project Euler, Reciprocal cycles: Problem 26
• Eric Weisstein's World of Mathematics, Repeating Decimal

Crossrefs

Cf. A051626, A007732.

Contains A001913.

Programs

Maple

count:= 1: A[1]:= 1: m:= 0:
for k from 0 to 100 do
  for d in [3,7,9,11] do
     x:= 10*k+d;
     p:= numtheory:-order(10,x);
     if p > m then
        m := p;
        count:= count+1;
        A[count]:= x
     fi
od od:
seq(A[i],i=1..count); # Robert Israel, Feb 10 2019

Mathematica

ResourceFunction["ProgressiveMaxPositions"]@
 Map[n |->
    First[RealDigits[n]] /. {{_, list_?ListQ} :> Length[list],
      list_?ListQ -> 0}][
  1/Range[1050]] (* Peter Cullen Burbery, Aug 05 2023 *)

Formula

RECORDS transform of A051626.