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A000188
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(1) Number of solutions to x^2 = 0 (mod n). (2) Also square root of largest square dividing n. (3) Also Max_{ d divides n } GCD[d,n/d].
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75
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1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 5, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 7, 5, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 5, 2, 1, 1, 1, 4, 9, 1, 1, 2, 1, 1, 1, 2, 1, 3
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OFFSET
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1,4
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COMMENTS
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Shadow transform of the squares A000290. - Vladeta Jovovic, Aug 02 2002
Labos Elemer (LABOS(AT)ana.sote.hu) and Henry Bottomley independently proved that (2) and (3) define the same sequence. Bottomley also showed that (1) and (2) define the same sequence.
Labos: Proof that (2)=(3): Let Max{[GCD[d,n/d]}=K, then d=Kx,n/d=Ky so n=KKxy where xy is the squarefree part of n,otherwise K is not maximal. Observe also that g=GCD[K,xy] is not necessarily 1. Thus K is also the "maximal square-root factor" of n.
A note on square roots of numbers: we can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n), LCM(b,c) = A007947(n) = "squarefree kernel" of n and bc = A019554(n) = "outer square root" of n.
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REFERENCES
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John M. Campbell, An Integral Representation of Kekule' Numbers, and Double Integrals Related to Smarandache Sequences, Arxiv preprint arXiv:1105.3399, 2011.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 1..10000
H. Bottomley, Some Smarandache-type multiplicative sequences
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, vol. 101, (2002), pp. 105-114.
S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n (arXiv:math.NT/0604465).
Gerry Myerson, Trifectas in Geometric Progression, Australian Mathematical Society Gazette, 35 (3) 2008, p 189-194
N. J. A. Sloane, Transforms
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FORMULA
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a(n) = Sum_{d^2|n} phi(d), where phi = Euler totient function A000010.
Multiplicative with a(p^e) = p^[e/2]. - David W. Wilson, Aug 01, 2001.
Dirichlet series: Sum(n=1..inf, a(n)/n^s) = zeta(2*s-1)*zeta(s)/zeta(2*s), (Re(s)>1).
Dirichlet convolution of A037213 and A008966. - R. J. Mathar, Feb 27 2011
Finch & Sebah show that the average order of a(n) is 3 log n/Pi^2. - Charles R Greathouse IV, Jan 03 2013
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MAPLE
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with(numtheory):A000188 := proc(n) local i: RETURN(op(mul(i, i=map(x->x[1]^floor(x[2]/2), ifactors(n)[2])))); end;
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MATHEMATICA
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Array[ Function[ n, Count[ Array[ PowerMod[ #, 2, n ]&, n, 0 ], 0 ] ], 100 ]
nn=90; slst=Range[Floor[Sqrt[nn]]]^2; Sqrt[#]&/@Table[Last[Select[ slst, Divisible[n, #]&]], {n, nn}] (* From Harvey P. Dale, May 11 2011 *)
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PROG
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(PARI) a(n)=if(n<1, 0, sum(i=1, n, i*i%n==0))
(PARI) a(n)=sqrtint(n/core(n)) [From Zak Seidov, Apr 07 2009]
(PARI) a(n)=core(n, 1)[2] \\ Michel Marcus, Feb 27 2013
(Haskell)
a000188 n = product $ zipWith (^)
(a027748_row n) $ map (`div` 2) (a124010_row n)
-- Reinhard Zumkeller, Apr 22 2012
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CROSSREFS
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a(n) = n/A019554(n).
Cf. A008833, A007913, A117811, A046951, A055210, A027748, A124010, A007913, A007947, A019554. For partial sums see A120486.
Sequence in context: A184957 A205792 A071773 * A162401 A191090 A097886
Adjacent sequences: A000185 A000186 A000187 * A000189 A000190 A000191
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KEYWORD
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nonn,easy,nice,mult
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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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