

A048798


Smallest k > 0 such that n*k is a perfect cube.


10



1, 4, 9, 2, 25, 36, 49, 1, 3, 100, 121, 18, 169, 196, 225, 4, 289, 12, 361, 50, 441, 484, 529, 9, 5, 676, 1, 98, 841, 900, 961, 2, 1089, 1156, 1225, 6, 1369, 1444, 1521, 25, 1681, 1764, 1849, 242, 75, 2116, 2209, 36, 7, 20, 2601, 338, 2809, 4, 3025, 49, 3249
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OFFSET

1,2


COMMENTS

Note that in general the smallest number k(>0) such that nk is a perfect mth power (rather obviously) = (the smallest mth power divisible by n)/n and also (slightly less obviously) =n^(m1)/(the number of solutions of x^m==0 mod n)^m.  Henry Bottomley, Mar 03 2000
Multiplicative with a(p^e) = p^(2  e%3).  Mitch Harris, May 17 2005


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Peter Kagey)
Krassimir T. Atanassov, On the 22nd, the 23rd and 24th Smarandache Problems, Notes on Number Theory and Discrete Mathematics, Sophia, Bulgaria, Vol. 5, No. 2 (1998), pp. 8082.
Krassimir T. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 1621.
Henry Bottomley, Some Smarandachetype multiplicative sequences.
Marcela Popescu and Mariana Nicolescu, About the Smarandache Complementary Cubic Function, Smarandache Notions Journal, Vol. 7, No. 123, 1996, pp. 5462.
Florentin Smarandache, Only Problems, Not Solutions!.


FORMULA

a(n) = A053149(n)/n = n^2/A000189(n)^3.


EXAMPLE

a(12) = a(2*2*3) = 2*3*3 = 18 since 12*18 = 6^3.
a(28) = a(2*2*7) = 2*7*7 = 98 since 28*98 = 14^3.


MATHEMATICA

a[n_] := For[k = 1, True, k++, If[ Divisible[c = k^3, n], Return[c/n]]]; Table[a[n], {n, 1, 60}] (* JeanFrançois Alcover, Sep 03 2012 *)
f[p_, e_] := p^(Mod[3  e, 3]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 10 2020 *)


PROG

(PARI) a(n)=my(f=factor(n)); prod(i=1, #f[, 1], f[i, 1]^(f[i, 2]%3)) \\ Charles R Greathouse IV, Feb 27 2013
(PARI) a(n)=for(k=1, n^2, if(ispower(k*n, 3), return(k)))
vector(100, n, a(n)) \\ Derek Orr, Feb 07 2015


CROSSREFS

Cf. A000189, A007913, A053149.
Cf. A254767 (analogous sequence with the restriction that k > n).
Sequence in context: A256513 A064505 A253288 * A007914 A048758 A277802
Adjacent sequences: A048795 A048796 A048797 * A048799 A048800 A048801


KEYWORD

nonn,easy,mult


AUTHOR

Charles T. Le (charlestle(AT)yahoo.com)


EXTENSIONS

More terms from Patrick De Geest, Feb 15 2000


STATUS

approved



