A080117 -id:A080117 - cp's OEIS frontend

A080118 Intersection of A080117 and A014486.

2, 10, 52, 738, 2866, 4023888, 921787428, 48034443442, 68055676507664, 210067141980993186, 1170565600000913519680, 257265188079961379006564, 3380147659553723806281906, 4190418227928183517574537416244

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

a(n) = A014486(A080119(n)). Same sequence in binary: A080120.

Maple
```
A080118 := n -> A080117(A080112(n));
```

A080110 Characteristic function of A080112.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

Index entries for characteristic functions

Maple
```
A080110 := n -> A080116(A080117(n));
```

def A080110(n) : return A080116(A080117(n)) # Peter Luschny, Aug 09 2012

A080111 Characteristic function of A080113.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

Index entries for characteristic functions

Maple
```
A080111 := n -> 1-A080116(A080117(n));
```

A080146 Binary encoding of quadratic residue set for each prime. a(n) = A055094(A000040(n)).

Original entry on oeis.org

1, 2, 9, 52, 738, 2829, 53643, 162438, 4023888, 166236537, 921787428, 48034254669, 935251837851, 2558696229078, 68055676507664, 2655011771373417, 210067141980993186, 831463105466530077, 42882922858578320598

Offset: 1

Antti Karttunen, Feb 11 2003

nonn

Cf. A000040, A055094, A080117, A080148.

with(numtheory,ithprime); A080146 := n -> A055094(ithprime(n));

A055094[n_] := With[{rr = Table[Mod[k^2, n], {k, 1, n-1}] // Union}, Boole[MemberQ[rr, #]]& /@ Range[n-1]] // FromDigits[#, 2]&;
a[n_] := A055094[Prime[n]];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Sep 20 2022 *)

a(n) = my(p=prime(n)); sum(k=1, p-1, 2^(k-1)*(0Michel Marcus, Sep 20 2022; after A055094

a(A080148(n)) = A080117(A080148(n)).

A080261 Simple involution of natural numbers: complement [binary_width(n)/2] least significant bits in the binary expansion of n.

Original entry on oeis.org

0, 1, 3, 2, 5, 4, 7, 6, 11, 10, 9, 8, 15, 14, 13, 12, 19, 18, 17, 16, 23, 22, 21, 20, 27, 26, 25, 24, 31, 30, 29, 28, 39, 38, 37, 36, 35, 34, 33, 32, 47, 46, 45, 44, 43, 42, 41, 40, 55, 54, 53, 52, 51, 50, 49, 48, 63, 62, 61, 60, 59, 58, 57, 56, 71, 70, 69, 68, 67, 66, 65, 64, 79

Offset: 0

Antti Karttunen, Feb 11 2003

nonn

			Binary expansion of 9 is 1001, we complement 4/2 = two rightmost bits, yielding 1010 = 10, thus a(9)=10. Binary expansion of 20 is 10100, we complement [5/2] = 2 rightmost bits, giving 10111 = 23, thus a(20)=23.

Used to construct A080117.

A080261 := proc(n) local w; w := floor(binwidth(n)/2); RETURN(((2^w)*floor(n/(2^w)))+(((2^w)-1)-ANDnos(n,(2^w)-1))); end;
binwidth := n -> (`if`((0 = n),1,floor_log_2(n)+1));
floor_log_2 := proc(n) local nn,i; nn := n; for i from -1 to n do if(0 = nn) then RETURN(i); fi; nn := floor(nn/2); od; end;

A080261[n_] := With[{w = Floor[binwidth[n]/2]}, 2^w* Floor[n/2^w] + ((2^w) - 1) - BitAnd[n, 2^w - 1]];
binwidth [n_] :=  If[0 == n, 1, floorLog2[n] + 1];
floorLog2[n_] := Module[{nn = n, i}, For[i = -1, i <= n, i++, If[0 == nn, Return[i]]; nn = Floor[nn/2]]];
Table[A080261[n], {n, 0, 100}] (* Jean-François Alcover, Sep 20 2022, after Maple program *)

a(n) = my(b=binary(n), k); if (#b%2, k=#b\2+2, k=#b/2+1); for (i=k, #b, b[i]=1-b[i]); fromdigits(b, 2); \\ Michel Marcus, Sep 20 2022

def A080261(n) :
    w = (2*n).exact_log(2) if n != 0 else 1
    w2 = 1 << w//2
    return w2*(n//w2) + w2 - 1 - (n&(w2-1))
[A080261(n) for n in (0..72)] # Peter Luschny, Aug 08 2012

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

A080118 Intersection of A080117 and A014486.

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Maple

A080110 Characteristic function of A080112.

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Sage

A080111 Characteristic function of A080113.

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A080146 Binary encoding of quadratic residue set for each prime. a(n) = A055094(A000040(n)).

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A080261 Simple involution of natural numbers: complement [binary_width(n)/2] least significant bits in the binary expansion of n.

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Maple

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Sage

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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