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A019554
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Smallest number whose square is divisible by n.
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23
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1, 2, 3, 2, 5, 6, 7, 4, 3, 10, 11, 6, 13, 14, 15, 4, 17, 6, 19, 10, 21, 22, 23, 12, 5, 26, 9, 14, 29, 30, 31, 8, 33, 34, 35, 6, 37, 38, 39, 20, 41, 42, 43, 22, 15, 46, 47, 12, 7, 10, 51, 26, 53, 18, 55, 28, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70, 71, 12, 73, 74, 15, 38, 77
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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
| F. Smarandache, "Collected Papers", Vol. II, Tempus Publ. Hse, Bucharest, 1996.
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LINKS
| 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.
F. Smarandache, Collected Papers, Vol. II
Eric Weisstein's World of Mathematics, Smarandache Ceil Function
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FORMULA
| Replace any square factors in n by their square roots.
Multiplicative with a(p^e) = p^ceil(e/2). Dirichlet series: Sum(n=1..inf, a(n)/n^s) = zeta(2*s-1)*zeta(s-1)/zeta(2*s-2), (Re(s)>2); Sum(n=1..inf, (1/a(n))/n^s) = zeta(2*s+1)*zeta(s+1)/zeta(2*s+2), (Re(s)>0)
a(n) = denominator of n/n^(3/2). [Arkadiusz Wesolowski, Dec 04 2011]
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MAPLE
| with(numtheory):A019554 := proc(n) local i: RETURN(op(mul(i, i=map(x->x[1]^ceil(x[2]/2), ifactors(n)[2])))); end;
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MATHEMATICA
| Flatten[Table[Select[Range[n], Divisible[#^2, n]&, 1], {n, 100}]] (* From Harvey P. Dale, Oct 17 2011 *)
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PROG
| (PARI) a(n)=n/core(n, 1)[2] \\ Charles R Greathouse IV, Feb 24, 2011
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CROSSREFS
| Cf. A019555, A008833, A015049.
a(n) = n/A000188(n)
Cf. A000188 A007913 A007947
Sequence in context: A062789 A066069 A019530 * A066729 A076685 A186646
Adjacent sequences: A019551 A019552 A019553 * A019555 A019556 A019557
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KEYWORD
| nonn,easy,mult,nice
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AUTHOR
| R. Muller
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