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A019554 Smallest number whose square is divisible by n. 25
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; text; internal format)
OFFSET

1,2

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.

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, Tempus Publ. Hse, Bucharest, 1996.

Eric Weisstein's World of Mathematics, Smarandache Ceil Function

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]

a(n) = product(A027748(n,k)^ceiling(a124010(n,k)/2): k=1..A001221(n)). - Reinhard Zumkeller, Apr 13 2013

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;

MATHEMATICA

Flatten[Table[Select[Range[n], Divisible[#^2, n]&, 1], {n, 100}]] (* Harvey P. Dale, Oct 17 2011 *)

PROG

(PARI) a(n)=n/core(n, 1)[2] \\ Charles R Greathouse IV, Feb 24, 2011

(Haskell)

a019554 n = product $ zipWith (^)

            (a027748_row n) (map ((`div` 2) . (+ 1)) $ a124010_row n)

-- Reinhard Zumkeller, Apr 13 2013

CROSSREFS

Cf. A019555, A008833, A015049, A000188 A007913 A007947.

a(n) = n/A000188(n)

Sequence in context: A062789 A066069 A019530 * A076685 A254503 A186646

Adjacent sequences:  A019551 A019552 A019553 * A019555 A019556 A019557

KEYWORD

nonn,easy,mult,nice

AUTHOR

R. Muller

STATUS

approved

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Last modified March 28 19:06 EDT 2017. Contains 284246 sequences.