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A000224 Number of squares mod n. 35
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

REFERENCES

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

LINKS

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

S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n (arXiv:math.NT/0604465).

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

FORMULA

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

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

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

seq(nops({seq(n^2 mod k, n=1..100)}), k=1..65); # Emeric Deutsch

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: # 2nd implementation, R. J. Mathar, Oct 10 2014

MATHEMATICA

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

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: A144000 A085202 A096009 * A085201 A051601 A193921

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 November 24 15:00 EST 2014. Contains 249899 sequences.