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A005356 Number of low discrepancy sequences in base 2.

%I A005356 M2435 #30 Mar 09 2023 08:04:53
%S A005356 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,
%T A005356 101,107,113,119,125,131,137,143,149,155,161,167,173,179,185,191,198,
%U A005356 205,212,219,226,233,240,247,254,261,268,275,282,289,296
%N A005356 Number of low discrepancy sequences in base 2.
%D A005356 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005356 Harald Niederreiter, <a href="http://dx.doi.org/10.1016/0022-314X(88)90025-X">Low-discrepancy and low-dispersion sequences</a>, J. Number Theory 30 (1988), no. 1, 51-70.
%p A005356 N := proc(b,n)
%p A005356     option remember;
%p A005356     local d;
%p A005356     add(b^d*numtheory[mobius](n/d),d=numtheory[divisors](n)) ;
%p A005356     %/n ;
%p A005356 end proc:
%p A005356 M := proc(b,n)
%p A005356     local h;
%p A005356     if n = 0 then
%p A005356         0;
%p A005356     else
%p A005356         add(N(b,h),h=1..n) ;
%p A005356     end if;
%p A005356 end proc:
%p A005356 nMax := proc(b,s)
%p A005356     local n;
%p A005356     for n from 0 do
%p A005356         if M(b,n) > s then
%p A005356             return n-1 ;
%p A005356         end if;
%p A005356     end do:
%p A005356 end proc:
%p A005356 A005356 := proc(s)
%p A005356     local n,b;
%p A005356     b := 2 ;
%p A005356     n := nMax(b,s) ;
%p A005356     n*(s-M(b,n))+add( (h-1)*N(b,h),h=1..n) ;
%p A005356 end proc:
%p A005356 seq(A005356(n),n=1..40) ; # _R. J. Mathar_, Jun 09 2016
%t A005356 Np[b_, n_] := Np[b, n] = Sum[b^d*MoebiusMu[n/d], {d, Divisors[n]}]/n;
%t A005356 M[b_, n_] := If[n == 0, 0, Sum[Np[b, h], {h, 1, n}]];
%t A005356 nMax[b_, s_] := Module[{n}, For[n = 0, True, n++, If[M[b, n] > s, Return[n - 1]]]];
%t A005356 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}]];
%t A005356 Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Mar 09 2023, after _R. J. Mathar_ *)
%Y A005356 Cf. A005357 (base 3), A005377 (base 4), A005358 (base 5).
%K A005356 nonn
%O A005356 1,4
%A A005356 _N. J. A. Sloane_, _Simon Plouffe_
%E A005356 More terms from _Sean A. Irvine_, May 27 2016