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A000224
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Number of squares mod n.
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30
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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
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listen;
history;
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internal format)
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OFFSET
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1,2
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REFERENCES
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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.
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LINKS
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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).
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FORMULA
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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.
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MAPLE
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seq(nops({seq(n^2 mod k, n=1..100)}), k=1..65); (E. Deutsch)
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MATHEMATICA
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Length[Union[#]]& /@ Table[Mod[k^2, n], {n, 65}, {k, n}] (* From Jean-François Alcover, Aug 30 2011 *)
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PROG
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(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
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CROSSREFS
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a(n)=A105612(n)+1.
Cf. A095972.
Sequence in context: A144000 A085202 A096009 * A085201 A051601 A193921
Adjacent sequences: A000221 A000222 A000223 * A000225 A000226 A000227
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KEYWORD
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nonn,easy,nice,mult,changed
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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