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A000224 Number of squares mod n. 49
1, 2, 2, 2, 3, 4, 4, 3, 4, 6, 6, 4, 7, 8, 6, 4, 9, 8, 10, 6, 8, 12, 12, 6, 11, 14, 11, 8, 15, 12, 16, 7, 12, 18, 12, 8, 19, 20, 14, 9, 21, 16, 22, 12, 12, 24, 24, 8, 22, 22, 18, 14, 27, 22, 18, 12, 20, 30, 30, 12, 31, 32, 16, 12, 21, 24, 34, 18, 24, 24, 36, 12 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget

S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.

S. Li, On the number of elements with maximal order in the multiplicative group modulo n, Acta Arithm. 86 (2) (1998) 113, see proof of theorem 2.1

E. J. F. Primrose, The number of quadratic residues mod m, Math. Gaz. v. 61 (1977) n. 415, 60-61.

W. D. Stangl, Counting Squares in Z_n, Math. Mag. 69 (1996) 285-289.

FORMULA

a(n) = A105612(n) + 1.

Multiplicative with a(p^e) = floor(p^e/6) + 2 if p = 2; floor(p^(e+1)/(2p + 2)) + 1 if p > 2. - David W. Wilson, Aug 01 2001

a(2^n) = A023105(n). a(3^n) = A039300(n). a(5^n) = A039302(n). a(7^n) = A039304(n). - R. J. Mathar, Sep 28 2017

EXAMPLE

The sequence of squares (A000290) modulo 10 reads 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1,... and this reduced sequence contains a(10) = 6 different values, {0,1,4,5,6,9}. - R. J. Mathar, Oct 10 2014

MAPLE

A000224 := proc(m)

    {seq( modp(b^2, m), b=0..m-1) };

    nops(%) ;

end proc: # Emeric Deutsch

# 2nd implementation

A000224 := proc(n)

    local a, ifs, f, p, e, c ;

    a := 1 ;

    ifs := ifactors(n)[2] ;

    for f in ifs do

        p := op(1, f) ;

        e := op(2, f) ;

        if p = 2 then

            if type(e, 'odd') then

                a := a*(2^(e-1)+5)/3 ;

            else

                a := a*(2^(e-1)+4)/3 ;

            end if;

        else

            if type(e, 'odd') then

                c := 2*p+1 ;

            else

                c := p+2 ;

            end if;

            a := a*(p^(e+1)+c)/2/(p+1) ;

        end if;

    end do:

    a ;

end proc: # R. J. Mathar, Oct 10 2014

MATHEMATICA

Length[Union[#]]& /@ Table[Mod[k^2, n], {n, 65}, {k, n}] (* Jean-François Alcover, Aug 30 2011 *)

a[2] = 2; a[n_] := a[n] = Switch[fi = FactorInteger[n], {{_, 1}}, (fi[[1, 1]] + 1)/2, {{2, _}}, 3/2 + 2^fi[[1, 2]]/6 + (-1)^(fi[[1, 2]]+1)/6, {{_, _}}, {p, k} = fi[[1]]; 3/4 + (p-1)*(-1)^(k+1)/(4*(p+1)) + p^(k+1)/(2*(p+1)), _, Times @@ Table[ a[Power @@ f], {f, fi}]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 09 2015 *)

PROG

(PARI) a(n) = local(v, i); v = vector(n, i, 0); for(i=0, floor(n/2), v[i^2%n+1] = 1); sum(i=1, n, v[i]) \\ Franklin T. Adams-Watters, Nov 05 2006

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f[, 1], if(f[i, 1]==2, 2^f[1, 2]\6+2, f[i, 1]^(f[i, 2]+1)\(2*f[i, 1]+2)+1)) \\ Charles R Greathouse IV, Jul 15 2011

(Haskell)

a000224 n = product $ zipWith f (a027748_row n) (a124010_row n) where

   f 2 e = 2 ^ e `div` 6 + 2

   f p e = p ^ (e + 1) `div` (2 * p + 2) + 1

-- Reinhard Zumkeller, Aug 01 2012

CROSSREFS

Cf. A095972, A046530 (cubic residues), A052273 (4th powers), A052274 (5th powers), A052275 (6th powers), A085310 (7th powers), A085311 (8th powers), A085312 (9th powers), A085313 (10th powers), A085314 (11th powers), A228849 (12th powers).

Sequence in context: A318816 A085202 A096009 * A085201 A300401 A051601

Adjacent sequences:  A000221 A000222 A000223 * A000225 A000226 A000227

KEYWORD

nonn,easy,nice,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified December 18 16:22 EST 2018. Contains 318229 sequences. (Running on oeis4.)