| Displaying 1-9 of 9 results found.
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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, 841, 900, 961, 2, 1089, 1156, 1225, 6, 1369, 1444, 1521, 25, 1681, 1764, 1849, 242, 75, 2116, 2209, 36, 7
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, 841, 900, 961, 2, 1089, 1156, 1225, 6, 1369, 1444, 1521, 25, 1681, 1764, 1849, 242, 75, 2116, 2209, 36, 7, 20
Smallest cube divisible by n.
+10
13
1, 8, 27, 8, 125, 216, 343, 8, 27, 1000, 1331, 216, 2197, 2744, 3375, 64, 4913, 216, 6859, 1000, 9261, 10648, 12167, 216, 125, 17576, 27, 2744, 24389, 27000, 29791, 64, 35937, 39304, 42875, 216, 50653, 54872, 59319, 1000, 68921, 74088, 79507, 10648
FORMULA
Multiplicative with a(p^e) = p^(e + ((3-e) mod 3)).
Sum_{n>=1} 1/a(n) = Product_{p prime} ((p^3+2)/(p^3-1)) = 1.655234386560802506... . (End)
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(9)/(4*zeta(3))) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A013667* A330596/(4* A002117) = 0.1559906... . - Amiram Eldar, Oct 27 2022
MATHEMATICA
a[n_] := For[k = 1, True, k++, If[ Divisible[c = k^3, n], Return[c]]]; Table[a[n], {n, 1, 44}] (* Jean-François Alcover, Sep 03 2012 *) f[p_, e_] := p^(e + Mod[3 - e, 3]); a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *) scdn[n_]:=Module[{c=Ceiling[Surd[n, 3]]}, While[!Divisible[c^3, n], c++]; c^3]; Array[scdn, 50] (* Harvey P. Dale, Jun 13 2020 *)
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^(f[i, 2] + (3-f[i, 2])%3)); } \\ Amiram Eldar, Oct 27 2022
n divided by largest cubefree factor of n.
+10
10
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 3, 1, 2, 1, 1, 1, 1, 1, 1, 1, 16, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 9, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2
FORMULA
Multiplicative with a(p^e) = p^max(e-2, 0). - Amiram Eldar, Sep 07 2020 Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^s - 1/p^(2*s-1) + 1/p^(2*s)). - Amiram Eldar, Dec 07 2023
MATHEMATICA
f[p_, e_] := p^Max[e-2, 0]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 07 2020 *)
CROSSREFS
Cf. A000189, A000578, A007948, A008834, A019555, A048798, A050985, A053149, A053150, A056551, A056552. See A003557 for squares and A062379 for 4th powers. Differs from A073753 for the first time at n=90, where a(90) = 1, while A073753(90) = 3.
Powerfree kernel of cubefree part of n.
+10
7
1, 2, 3, 2, 5, 6, 7, 1, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 3, 5, 26, 1, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 5, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 2, 55, 7, 57, 58, 59, 30, 61, 62, 21, 1, 65, 66, 67, 34, 69, 70, 71, 3, 73, 74, 15, 38, 77
FORMULA
If n = Product_{j} Pj^Ej then a(n) = Product_{j} Pj^Fj, where Fj = 0 if Ej is 0 or a multiple of 3 and Fj = 1 otherwise.
Multiplicative with a(p^e) = p^(if 3|e, then 0, else 1). - Mitch Harris, Apr 19 2005 Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(6)/2) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4) = 0.3480772773... . - Amiram Eldar, Oct 28 2022 Dirichlet g.f.: zeta(3*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-1)). - Amiram Eldar, Sep 16 2023
EXAMPLE
a(32) = 2 because cubefree part of 32 is 4 and powerfree kernel of 4 is 2.
MATHEMATICA
f[p_, e_] := p^If[Divisible[e, 3], 0, 1]; a[n_] := Times @@ (f @@@ FactorInteger[ n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)
PROG
(PARI) a(n) = my(f=factor(n)); for (k=1, #f~, if (frac(f[k, 2]/3), f[k, 2] = 1, f[k, 2] = 0)); factorback(f); \\ Michel Marcus, Feb 28 2019
CROSSREFS
Cf. A000189, A000578, A007947, A008834, A013664, A019555, A048798, A050985, A053149, A053150, A056551.
Smallest cube divisible by n divided by largest cube which divides n.
+10
5
1, 8, 27, 8, 125, 216, 343, 1, 27, 1000, 1331, 216, 2197, 2744, 3375, 8, 4913, 216, 6859, 1000, 9261, 10648, 12167, 27, 125, 17576, 1, 2744, 24389, 27000, 29791, 8, 35937, 39304, 42875, 216, 50653, 54872, 59319, 125, 68921, 74088, 79507, 10648
FORMULA
Multiplicative with a(p^e) = 1 if e is divisible by 3, and a(p^e) = p^3 otherwise.
Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(12)/(4*zeta(3))) * Product_{p prime} (1 - 1/p^2 + 1/p^3) = A013670 * A330596 / (4* A002117) = 0.1557163105... . (End) Dirichlet g.f.: zeta(3*s) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(2*s-3)). - Amiram Eldar, Sep 16 2023
EXAMPLE
a(16) = 8 since smallest cube divisible by 16 is 64 and smallest cube which divides 16 is 8 and 64/8 = 8.
MATHEMATICA
f[p_, e_] := p^If[Divisible[e, 3], 0, 1]; a[n_] := (Times @@ (f @@@ FactorInteger[ n]))^3; Array[a, 100] (* Amiram Eldar, Aug 29 2019*)
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%3, f[i, 1], 1))^3; } \\ Amiram Eldar, Oct 28 2022
Smallest number k (k>0) such that n*k is a perfect 4th power.
+10
4
1, 8, 27, 4, 125, 216, 343, 2, 9, 1000, 1331, 108, 2197, 2744, 3375, 1, 4913, 72, 6859, 500, 9261, 10648, 12167, 54, 25, 17576, 3, 1372, 24389, 27000, 29791, 8, 35937, 39304, 42875, 36, 50653, 54872, 59319, 250, 68921, 74088, 79507, 5324, 1125
FORMULA
Multiplicative with a(p^e) = p^((4 - e) mod 4). - Amiram Eldar, Sep 08 2020 Sum_{k=1..n} a(k) ~ c * n^4, where c = (zeta(16)/(4*zeta(4))) * Product_{p prime} (1 - 1/p^2 + 1/p^4 - 1/p^7 + 1/p^8) = 0.1537848996... . - Amiram Eldar, Oct 27 2022
EXAMPLE
a(64) = 4 because the smallest 4th power divisible by 64 is 256 and 64*4 = 256.
MATHEMATICA
f[p_, e_] := p^Mod[4 - e, 4]; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 08 2020 *)
a(n) is the least k > n such that k*n is a cube.
+10
4
8, 4, 9, 16, 25, 36, 49, 27, 24, 100, 121, 18, 169, 196, 225, 32, 289, 96, 361, 50, 441, 484, 529, 72, 40, 676, 64, 98, 841, 900, 961, 54, 1089, 1156, 1225, 48, 1369, 1444, 1521, 200, 1681, 1764, 1849, 242, 75, 2116, 2209, 288, 56, 160, 2601, 338, 2809, 108
COMMENTS
a(n) <= n^2 for all n > 1 because n * n^2 = n^3.
EXAMPLE
a(12) = 18 because 12*18 = 6^3 (and 12*13, 12*14, 12*15, 12*16, 12*17 are not perfect cubes).
MATHEMATICA
f[n_] := Block[{k = n + 1}, While[! IntegerQ@ Power[k n, 1/3], k++]; k]; Array[f, 54] (* Michael De Vlieger, Mar 17 2015 *)
PROG
(Ruby)
def a(n)
min = (n**(2/3.0)).ceil
(min..n+1).each { |i| return i**3/n if i**3 % n == 0 && i**3 > n**2 }
end
(PARI) a(n)=if (n==1, 8, for(k=n+1, n^2, if(ispower(k*n, 3), return(k))))
vector(100, n, a(n))) \\ Derek Orr, Feb 07 2015 (PARI) a(n) = {f = factor(n); for (i=1, #f~, if (f[i, 2] % 3, f[i, 2] = 3 - f[i, 2]); ); cb = factorback(f); cbr = sqrtnint(cb*n, 3); cb = cbr^3; k = cb/n; while((type(k=cb/n) != "t_INT") || (k<=n), cbr++; cb = cbr^3; ); k; } \\ Michel Marcus, Mar 14 2015
CROSSREFS
Cf. A072905 (an analogous sequence for squares). Cf. A048798 (similar sequence, no restriction that a(n) > n).
The least k > 0 such that k* A004709(n) is a cube.
+10
2
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
COMMENTS
This is a permutation of the cubefree numbers ( A004709).
FORMULA
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
EXAMPLE
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
MAPLE
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:
MATHEMATICA
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 *)
PROG
(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 (Python)
from math import prod
from sympy import mobius, factorint, integer_nthroot
def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1))
m, k = n, f(n)
while m != k:
m, k = k, f(k)
return prod(p**(-e%3) for p, e in factorint(m).items()) # Chai Wah Wu, Aug 05 2024
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