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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. [The relation with LCM is wrong if b is not squarefree. One must, e.g., replace b with A007947(b). - M. F. Hasler, Mar 03 2018] 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^ceiling(e/2). Dirichlet series:    Sum_{n>=1} a(n)/n^s = zeta(2*s-1)*zeta(s-1)/zeta(2*s-2), (Re(s)>2);    Sum_{n>=1} (1/a(n))/n^s = zeta(2*s+1)*zeta(s+1)/zeta(2*s+2), (Re(s)>0). a(n) = n/A000188(n). a(n) = denominator of n/n^(3/2). - Arkadiusz Wesolowski, Dec 04 2011 a(n) = Product_{k=1..A001221(n)} A027748(n,k)^ceiling(a124010(n,k)/2). - Reinhard Zumkeller, Apr 13 2013 MAPLE with(numtheory):A019554 := proc(n) local i: RETURN(op(mul(i, i=map(x->x^ceil(x/2), ifactors(n))))); 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) \\ 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. A000188, A007913, A007947, A008833, A015049, A019555. 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 September 15 22:10 EDT 2019. Contains 327088 sequences. (Running on oeis4.)