A055095 -id:A055095 - cp's OEIS frontend

A055131 Those composite s for which A055095[s] = 2.

15, 39, 51, 87, 111, 123, 159, 183, 219, 267, 291, 303, 327, 339, 411, 447, 471, 519, 543, 579, 591, 687, 699, 723, 771, 807, 831, 843, 879, 939, 951, 1011, 1047, 1059, 1119, 1167, 1191, 1203, 1227, 1263, 1299, 1347, 1371, 1383, 1527, 1563, 1623, 1671

Offset: 0

Antti Karttunen, Apr 04 2000

nonn

find_A055095_is_2_composites := proc(upto_n) local j,a; a := []; for j from 1 to upto_n do if(-1 = (j - wt(GrayCode(qrs2bincode((2*j)+1))))) then if(not isprime((2*j)+1)) then a := [op(a),((2*j)+1)]; fi; fi; od; RETURN(a); end;

A005811[n_] := Length[Length /@ Split[IntegerDigits[n, 2]]];
A055094[n_] := With[{rr = Table[Mod[k^2, n], {k, 1, n-1}] // Union}, Boole[MemberQ[rr, #]] & /@ Range[n-1]] // FromDigits[#, 2]&;
A055095[1] = 0; A055095[n_] := 2*A005811[A055094[n]] - (n-1);
A055131 = Position[Array[A055095, 2000], 2] // Flatten // Select[#, CompositeQ]& (* Jean-François Alcover, Mar 06 2016 *)

a(n) = 3*((4*A005098[n])+1) = 3*A002144[n] ??? (Conjecture, not yet proved)

More terms from James Sellers, Apr 21 2000

A055094 Binary encoding of quadratic residue set of n. L(1/n) is the most significant bit, L(n-1/n) is the least significant bit, i.e., the rows of A055088 interpreted as binary numbers.

Original entry on oeis.org

0, 1, 2, 4, 9, 22, 52, 72, 146, 313, 738, 1156, 2829, 6772, 9520, 18496, 53643, 75154, 162438, 312328, 600116, 1513186, 4023888, 4737152, 9741609, 23182093, 38478994, 76286020, 166236537, 311977264, 921787428, 1212203072, 2962424994

Offset: 1

Antti Karttunen, Apr 04 2000

nonn

L(a/n) stands for generalized Legendre symbol, with value = 1 only if a is a quadratic residue of n.

Cf. A055088, A054432, A055095.

A055094 := proc(n)
    local i, z;
    z := 0;
    for i from 1 to n-1 do
        z := z*2;
        if (1 = numtheory[quadres](i, n)) then
            z := z + 1;
        fi;
    od;
    return z;
end proc:

a[n_] := With[{rr = Table[Mod[k^2, n], {k, 1, n - 1}] // Union}, Boole[ MemberQ[rr, #]]& /@ Range[n - 1]] // FromDigits[#, 2]&; Array[a, 40] (* Jean-François Alcover, Mar 05 2016*)

{a(n)=sum(k=1, n-1, 2^(k-1)*(0Michael Somos, Oct 14 2006 */

def A055094(n) :
    Q = quadratic_residues(n)
    z = 0
    for i in (1..n-1)  :
        z = z*2
        if i in Q : z += 1
    return z
[A055094(n) for n in (1..33)] # Peter Luschny, Aug 08 2012

a(n) = qrs2bincode(n)

cp's OEIS Frontend

A055131 Those composite s for which A055095[s] = 2.

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A055094 Binary encoding of quadratic residue set of n. L(1/n) is the most significant bit, L(n-1/n) is the least significant bit, i.e., the rows of A055088 interpreted as binary numbers.

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