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A005356
Number of low discrepancy sequences in base 2.
4
0, 0, 1, 3, 5, 8, 11, 14, 18, 22, 26, 30, 34, 38, 43, 48, 53, 58, 63, 68, 73, 78, 83, 89, 95, 101, 107, 113, 119, 125, 131, 137, 143, 149, 155, 161, 167, 173, 179, 185, 191, 198, 205, 212, 219, 226, 233, 240, 247, 254, 261, 268, 275, 282, 289, 296
OFFSET
1,4
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Harald Niederreiter, Low-discrepancy and low-dispersion sequences, J. Number Theory 30 (1988), no. 1, 51-70.
MAPLE
N := proc(b, n)
option remember;
local d;
add(b^d*numtheory[mobius](n/d), d=numtheory[divisors](n)) ;
%/n ;
end proc:
M := proc(b, n)
local h;
if n = 0 then
0;
else
add(N(b, h), h=1..n) ;
end if;
end proc:
nMax := proc(b, s)
local n;
for n from 0 do
if M(b, n) > s then
return n-1 ;
end if;
end do:
end proc:
A005356 := proc(s)
local n, b;
b := 2 ;
n := nMax(b, s) ;
n*(s-M(b, n))+add( (h-1)*N(b, h), h=1..n) ;
end proc:
seq(A005356(n), n=1..40) ; # R. J. Mathar, Jun 09 2016
MATHEMATICA
Np[b_, n_] := Np[b, n] = Sum[b^d*MoebiusMu[n/d], {d, Divisors[n]}]/n;
M[b_, n_] := If[n == 0, 0, Sum[Np[b, h], {h, 1, n}]];
nMax[b_, s_] := Module[{n}, For[n = 0, True, n++, If[M[b, n] > s, Return[n - 1]]]];
a[s_] := Module[{n, b}, b = 2; n = nMax[b, s]; n*(s - M[b, n]) + Sum[(h - 1)*Np[b, h], {h, 1, n}]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Mar 09 2023, after R. J. Mathar *)
CROSSREFS
Cf. A005357 (base 3), A005377 (base 4), A005358 (base 5).
Sequence in context: A052488 A076372 A248611 * A060432 A156023 A261223
KEYWORD
nonn
EXTENSIONS
More terms from Sean A. Irvine, May 27 2016
STATUS
approved