OFFSET
1,2
COMMENTS
Note that in general the smallest number k(>0) such that nk is a perfect m-th power (rather obviously) = (the smallest m-th power divisible by n)/n and also (slightly less obviously) =n^(m-1)/(the number of solutions of x^m==0 mod n)^m. - Henry Bottomley, Mar 03 2000
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. 80-82.
Krassimir T. Atanassov, On Some of Smarandache's Problems, American Research Press, 1999, 16-21.
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Marcela Popescu and Mariana Nicolescu, About the Smarandache Complementary Cubic Function, Smarandache Notions Journal, Vol. 7, No. 1-2-3, 1996, pp. 54-62.
Florentin Smarandache, Only Problems, Not Solutions! (see Unsolved Problem: 28, p. 26).
FORMULA
Multiplicative with a(p^e) = p^((-e) mod 3). - Mitch Harris, May 17 2005
Sum_{k=1..n} a(k) ~ c * n^3, where c = (zeta(9)/(3*zeta(3)) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = 0.2079875504... . - Amiram Eldar, Oct 28 2022
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}] (* Jean-François Alcover, Sep 03 2012 *)
f[p_, e_] := p^(Mod[-e, 3]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 10 2020 *)
With[{cbs=Range[3300]^3}, Table[SelectFirst[cbs, Mod[#, n]==0&]/n, {n, 60}]] (* Harvey P. Dale, May 10 2024 *)
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
(Python)
from math import prod
from sympy import factorint
def A048798(n): return prod(p**(-e%3) for p, e in factorint(n).items()) # Chai Wah Wu, Aug 05 2024
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Charles T. Le (charlestle(AT)yahoo.com)
EXTENSIONS
More terms from Patrick De Geest, Feb 15 2000
STATUS
approved