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A048798
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Smallest k > 0 such that n*k is a perfect cube.
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10
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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
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1,2
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COMMENTS
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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
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A254767 (analogous sequence with the restriction that k > n).
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KEYWORD
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nonn,easy,mult
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AUTHOR
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Charles T. Le (charlestle(AT)yahoo.com)
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EXTENSIONS
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STATUS
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approved
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