%I #18 Sep 20 2022 07:54:30

%S 2,10,52,738,2866,53620,162438,4023888,166243974,921787428,

%T 48034443442,935251508324,2558696229078,68055676507664,

%U 2655011787909270,210067141980993186,831463106366605026,42882922858578320598

%N Binary encoding of quadratic residue set formed for n-th prime, coerced to "complementarily symmetric binary sequence" with A080261 if the prime is of the form 4k+1.

%F a(A080148(n)) = A080146(A080148(n))

%p with(numtheory,ithprime); A080117 := proc(n) local c,p; p := ithprime(n); c := A055094(p); if(3 = (p mod 4)) then RETURN(c); else RETURN(A080261(c)); fi; end;

%t A055094[n_] := With[{rr = Table[Mod[k^2, n], {k, 1, n-1}] // Union}, Boole[ MemberQ[rr, #]] & /@ Range[n-1]] // FromDigits[#, 2]&;

%t A080261[n_] := Module[{bb = IntegerDigits[n, 2]}, lg = Length[bb]; Do[ bb[[i]] = 1 - bb[[i]], {i, lg, lg - Floor[lg/2] + 1, -1}]; FromDigits[ bb, 2]];

%t a[n_] := Module[{c, p = Prime[n]}, c = A055094[p]; If[Mod[p, 4] == 3, c, A080261[c]]]; Table[a[n], {n, 2, 20}] (* _Jean-François Alcover_, Mar 05 2016, adapted from Maple *)

%o (Sage) # uses[A080261]

%o def A080117(n) :

%o p = nth_prime(n)

%o c = A055094(p)

%o return c if 3 == p%4 else A080261(c)

%o [A080117(n) for n in (2..19)] # _Peter Luschny_, Aug 09 2012

%Y Cf. A080118, A080146, A080148.

%K nonn

%O 2,1

%A _Antti Karttunen_, Feb 11 2003