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Search: id:A000188
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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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68
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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
(list; graph; listen)
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OFFSET
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1,4
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COMMENT
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Shadow transform of the squares A000290. - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 02 2002
Labos Elemer (LABOS(AT)ana.sote.hu) and Henry Bottomley (se16(AT)btinternet.com) 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 square-free 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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Gerry Myerson, Trifectas in Geometric Progression, Australian Mathematical Society Gazette, 2008, to appear.
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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).
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 (davidwwilson(AT)comcast.net), 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).
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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 ]
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PROGRAM
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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 (zakseidov(AT)yahoo.com), Apr 07 2009]
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CROSSREFS
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a(n) = n/A019554(n)
Cf. A008833, A007913, A117811, A046951, A055210.
Cf. A007913, A007947, A019554. For partial sums see A120486.
Sequence in context: A046854 A066170 A071773 this_sequence A162401 A097886 A088863
Adjacent sequences: A000185 A000186 A000187 this_sequence 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 (njas(AT)research.att.com).
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