A080146 -id:A080146 - cp's OEIS frontend

A080117 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.

2, 10, 52, 738, 2866, 53620, 162438, 4023888, 166243974, 921787428, 48034443442, 935251508324, 2558696229078, 68055676507664, 2655011787909270, 210067141980993186, 831463106366605026, 42882922858578320598

Offset: 2

Antti Karttunen, Feb 11 2003

nonn

Cf. A080118, A080146, A080148.

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;

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

# uses[A080261]
def A080117(n) :
    p = nth_prime(n)
    c = A055094(p)
    return c if 3 == p%4 else A080261(c)
[A080117(n) for n in (2..19)] # Peter Luschny, Aug 09 2012

a(A080148(n)) = A080146(A080148(n))

A280109 a(n) is the decimal value corresponding to the binary representation of the distribution of quadratic residues (value=1) and non-quadratic residues (value=0) mod n, where numbers are ordered left to right from 0 to n-1.

Original entry on oeis.org

1, 3, 6, 12, 25, 54, 116, 200, 402, 825, 1762, 3204, 6925, 14964, 25904, 51264, 119179, 206226, 424582, 836616, 1648692, 3610338, 8218192, 13125760, 26518825, 56736525, 105587858, 210503748, 434671993, 848848176, 1995529252, 3359686720, 7257392290, 15621149067

Offset: 1

Adnan Baysal, Dec 26 2016

Sort mod n numbers {0,1,...,n-1} in ascending order. For each modular number i, write 1 if i is a quadratic residue mod n (i.e., it has a square root), else write 0. The corresponding n-bit number is a(n).

			For n = 10, quadratic residues are 0, 1, 4, 5, 6, 9 so a(10) is 1100111001 in binary which is 825.

Adnan Baysal, Table of n, a(n) for n = 1..1000

Cf. A055094, A080117, A080146, A096008.

f[n_] := Total[ 2^(n -1 -Union[ Mod[ Range[0, n - 1]^2, n]] )]; Array[f, 34] (* Robert G. Wilson v, Dec 28 2016 *)

def qr_distribution(N):
    QR = []
    QN = []
    for i in range(N):
        t = (i*i)%N
        if t not in QR: QR.append(t)
    for i in range(N):
        if i not in QR: QN.append(i)
    out = 0
    for i in range(0,N):
        out *= 2
        if i in QR: out += 1
    return out

cp's OEIS Frontend

A080117 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.

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A280109 a(n) is the decimal value corresponding to the binary representation of the distribution of quadratic residues (value=1) and non-quadratic residues (value=0) mod n, where numbers are ordered left to right from 0 to n-1.

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