

A019554


Smallest number whose square is divisible by n.


28



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]
Instead of the terms "inner square root" and "outer square root", we may use the terms "lower square root" and "upper square root", respectively. Upper kth roots have been studied by Broughan (2002, 2003, 2006).  Petros Hadjicostas, Sep 15 2019


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
H. Bottomley, Some Smarandachetype multiplicative sequences.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105114.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105114.
Kevin A. Broughan, Relationship between the integer conductor and kth root functions, Int. J. Pure Appl. Math. 5(3) (2003), 253275.
Kevin A. Broughan, Relaxations of the ABC Conjecture using integer k'th roots, New Zealand J. Math. 35(2) (2006), 121136.
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*s1)*zeta(s1)/zeta(2*s2), (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[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. A000188 (inner square root), A053150 (inner 3rd root), A019555 (outer 3rd root), A053164 (inner 4th root), A053166 (outer 4th root), A015052 (outer 5th root), A015053 (outer 6th root).
Cf. A007913, A007947, A008833, A015049.
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



