A005356 Number of low discrepancy sequences in base 2. (Formerly M2435)

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

Table of n, a(n) for n=1..56.

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

Adjacent sequences: A005353 A005354 A005355 * A005357 A005358 A005359

Keyword

nonn

Author

N. J. A. Sloane, Simon Plouffe

Extensions

More terms from Sean A. Irvine, May 27 2016

Status

approved