A102032 a(n) is the smallest positive integer k such that, if kn is written in base 2, it requires exactly n ones.

1, 3, 7, 15, 11, 21, 89, 255, 167, 307, 349, 1365, 1259, 6729, 6417, 65535, 28431, 29127, 54757, 209715, 274627, 750685, 706649, 5592405, 2663383, 9679163, 14913005, 186946121, 37025579, 353440017, 1175487521, 4294967295

Offset

1,2

Comments

This sequence can be considered for any base b. If one calculates the arithmetic mean of the sequence d(n):=a(n)/2^n, i.e. (d(1)+d(2)+...+d(n))/n, one obtains a sequence converging to zero.

From Robert Israel, Aug 26 2015: (Start)

a(2*n) <= (2^A070939(a(n)) + 1)*a(n).

if n is odd, a(n) <= (2^(n*r)-1)/(n*(2^r-1)) where r = A002326((n-1)/2). (End)

References

I. Vardi, Niven numbers, Computational Recreations in Mathematics, Addison-Wesley, 1991, pp. 19 and 28--31.

Links

Eugen J. Ionascu and Ray Chandler, Table of n, a(n) for n = 1..2024

J. M. De Koninck, N. Doyon and I. Katai, On the counting function for the Niven numbers, Acta Arith. 106 (2003), 265--275.

E. J. Ionascu, H. Fredricksen, F. Luca and P. Stanica, Minimal Niven numbers, Acta Arith. 132 (2008), 135--159.

E. J. Ionascu, F. Luca, P. Stanica and H. Fredricksen, Remarks on a sequence of minimal Niven numbers, Proceedings of the International Workshop, SSC 2007 (Sequences, Subsequences and Consequences) Springer, 162--168.

Formula

a(n) = 2^n-1 if n=2^k or a(n) = (2^(n+k-1)+2^n-2^(n-k)-1)/n if n=2^k-1 is a prime number; unknown for other values of n.

Example

Example: If n=7 then 7(89)=623 which written in base 2 is 1001101111 using exactly 7 ones and 89 is the smallest positive integer with this property. Hence a(7)=89. The number 1001101111 is usually known as Niven number in base 2. We called 623 a minimal Niven number.

Maple

with(numtheory):
fjv6:=proc(n, m)
local i, j, k, l, x, x1, y, y1, z, z1, w, stopp, s, t, u, v, A, F, G, out;
i:=n; stopp:=0;
x1:=2^(m*i+6)-1; x:=x1 mod i; j:=0;
while stopp=0 and j<=m*i+5 do
l:=j;
while stopp=0 and l<=m*i+4 do
k:=l;
while stopp=0 and k<=m*i+3 do
s:=k;
while stopp=0 and s<=m*i+2 do
t:=s;
while stopp=0 and t<=m*i+1 do
v:=t;
while stopp=0 and v<=m*i do y1:=2^(m*i+5-j)+2^(m*i+4-l)+2^(m*i+3-k)+2^(m*i+2-s)+2^(m*i+1-t)+2^(m*i-v); y:=y1 mod i;
if y=x then z:=(x1-y1)/i; out:=[m*i, z];
stopp:=1;
fi;
v:=v+1; od; t:=t+1; od; s:=s+1; od; k:=k+1; od; l:=l+1; od; j:=j+1; od;
if stopp=0 then out:=[m*i, 0]; fi;
out;
end:
formula:=proc(n)
local x, y, B, expon, outputis, theOddFactor;
x:=n+1; B:=ifactors(x); expon:=B[2][1][2]; theOddFactor:=(n+1)/2^expon;
y:=isprime(n);
if theOddFactor=1 and y=true then outputis:=[n, (2^(n+expon-1)+2^n-2^(n-expon)-1)/n]; fi;
if theOddFactor>1 or y=false then outputis:=fjv6(n, 1); fi;
lprint(outputis[1], outputis[2]);
end:
fjfromis6:=proc(n, m)
local k, B, expon, theoddfac, par, stopp, av, sub;
av:=0; for k from n to m do
par:=k mod 2;
if par=0 then B:=ifactors(k); expon:=B[2][1][2]; theoddfac:=k/2^expon;
sub:=fjv6(theoddfac, 2^expon);
lprint(sub[1], sub[2]); fi;
stopp:=0;
if par=1 then formula(k); fi;
od;
end:
fjfromis6(1, 185);
# Alternative:
F:= proc(k, x, n, dmax)
   option remember;
   local d, z, v;
   if k = 0 then
      if x = 0 then return 0 else return infinity fi
   end;
   for d from k-1 to dmax do
      v:= procname(k-1, (x - 2^d) mod n, n, d-1) ;
      if v < 2^d then return v + 2^d fi
   od;
   infinity;
end proc:
seq(F(n, 0, n, infinity)/n, n=1..100); # Robert Israel, Aug 26 2015

Mathematica

F[k_, x_, n_, dMax_] := F[k, x, n, dMax] = Module[{d, z, v}, If[k == 0, If[x == 0, Return[0], Return[Infinity]]]; For[d = k - 1, d <= dMax, d++, v = F[k - 1, Mod[x - 2^d, n], n, d - 1]; If[v < 2^d, Return[v + 2^d]]]; Infinity];
Table[F[n, 0, n, Infinity]/n, {n, 1, 32}] (* Jean-François Alcover, Jun 22 2020, after Robert Israel *)

Prog

(PARI) a(n)=my(K=n); while(hammingweight(K)!=n, K+=n); K/n \\ Charles R Greathouse IV, Feb 04 2013

Crossrefs

Cf. A005349, A143115.

Sequence in context: A348483 A333556 A334782 * A086517 A346296 A332374

Adjacent sequences: A102029 A102030 A102031 * A102033 A102034 A102035

Keyword

nonn

Author

Eugen J. Ionascu, Aug 03 2007

Extensions

Edited by Ray Chandler, Nov 16 2008

Status

approved