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A277802
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The least k > 0 such that k*A004709(n) is a cube.
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2
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1, 4, 9, 2, 25, 36, 49, 3, 100, 121, 18, 169, 196, 225, 289, 12, 361, 50, 441, 484, 529, 5, 676, 98, 841, 900, 961, 1089, 1156, 1225, 6, 1369, 1444, 1521, 1681, 1764, 1849, 242, 75, 2116, 2209, 7, 20, 2601, 338, 2809, 3025, 3249, 3364, 3481, 450, 3721, 3844
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OFFSET
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1,2
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COMMENTS
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This is a permutation of the cubefree numbers (A004709).
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LINKS
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FORMULA
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Sum_{k=1..n} a(k) ~ c * zeta(3)^3 * n^3 / 3, where c = Product_{p prime} (1 - 1/p^2 + 1/p^5 - 1/p^6) = 0.36052971192705404983... . - Amiram Eldar, Feb 20 2024
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EXAMPLE
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a(8) = 3 because 3 * A004709(8) = 3 * 9 = 3^3.
a(16) = 12 because A004709(16) = 18 = 2^1 * 3^2. The least k such that k * 2^1 * 3^2 is a cube is 2^(3 - (1 mod 3)) * 3^(3 - (2 mod 3)) = 12. - David A. Corneth, Nov 01 2016
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MAPLE
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f:= proc(n) local F, E;
F:= ifactors(n)[2];
E:= F[.., 2];
if max(E) >= 3 then return NULL fi;
mul(F[i, 1]^(3-E[i]), i=1..nops(F));
end proc:
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MATHEMATICA
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Table[k = 1; While[! IntegerQ[(k #)^(1/3)], k++] &@ #[[n]]; k, {n, 53}] &@ Select[Range[10^4], FreeQ[FactorInteger@ #, {_, k_ /; k > 2}] &] (* Michael De Vlieger, Nov 01 2016, after Jan Mangaldan at A004709 *)
f[p_, e_] := If[e > 2, 0, p^(Mod[-e, 3])]; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Array[s, 100], # > 0 &] (* Amiram Eldar, Feb 20 2024 *)
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PROG
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(PARI) \\ A list of about n terms (a little more probably).
lista(n) = {n = ceil(1.21*n); my(l=List([1]), f); forprime(p=2, n, for(i=1, #l, if(l[i] * p<=n, listput(l, l[i]*p); if(l[i]*p^2<=n, listput(l, l[i]*p^2))))); listsort(l); for(i=2, #l, f=factor(l[i]); f[, 2] = vector(#f[, 2], i, 3-(f[i, 2] % 3))~; l[i] = factorback(f)); l} \\ David A. Corneth, Nov 01 2016
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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