A178495 Numbers k whose sum of digits equals the period of 1/k.

1, 11, 41, 81, 111, 187, 231, 451, 567, 891, 1111, 1197, 1221, 1539, 1717, 1881, 2329, 2877, 3003, 3591, 3813, 4551, 4773, 5643, 6237, 6327, 7371, 8991, 9399, 9449, 10773, 11111, 11583, 13167, 13651, 15561, 16863, 17589, 23331, 24453, 27371, 39501

Offset: 1

Author

Michel Lagneau, May 28 2010

Keywords

• nonn
• base

Examples

			11 is in the sequence because 1 + 1 = 2 and 1/11 = 0.09 09 09... with period 2.
41 is in the sequence because 4 + 1 = 5 and 1/41 = 0.02439 02439 ... with period 5.
		

References

• J. W. L. Glaisher, On circulating decimals, Proc. Camb. Phil. Soc., 3 (1878), pp. 185-206.
• D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-12.

Links

• Eric Weisstein's World of Mathematics, Decimal Expansion.

Crossrefs

Cf. A002329, A007953, A084680.

Programs

Maple

with(numtheory): nn:= 2343: T:=array(1..nn): U:=array(1..nn): X:=array(1..nn): Y:=array(1..nn):k:=1:for n from 2 to 999999 do:for p from 1 to 55 while(irem(10^p,n)<>1 or gcd(n,10)<> 1) do: od: if irem(10^p,n) = 1 and gcd(n,10) = 1 then T[k]:=n : U[k]:=p: k:=k+1:else fi:od: for n from 1 to nn do:p1:= T[n]: l:=length(p1):n0:=p1:s:=0:for m from 1 to l do:q:=n0:u:=irem(q,10):v:=iquo(q,10):n0:=v :s:=s+u:od: X[n]:=s: od: for n from 1 to nn do:if X[n]= U[n] then printf(`%d, `, T[n]):else fi:od:

Formula

{k: A084680(k) = A007953(k)}. - R. J. Mathar, Jun 07 2010