A277802 - OEIS

%I #35 Aug 05 2024 17:35:55

%S 1,4,9,2,25,36,49,3,100,121,18,169,196,225,289,12,361,50,441,484,529,

%T 5,676,98,841,900,961,1089,1156,1225,6,1369,1444,1521,1681,1764,1849,

%U 242,75,2116,2209,7,20,2601,338,2809,3025,3249,3364,3481,450,3721,3844

%N The least k > 0 such that k*A004709(n) is a cube.

%C This is a permutation of the cubefree numbers (A004709).

%C a(n) <= A004709(n)^2, with equality iff A004709(n) is squarefree. - _Robert Israel_, Nov 09 2016

%H Peter Kagey, <a href="/A277802/b277802.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A048798(A004709(n)).

%F 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

%e a(8) = 3 because 3 * A004709(8) = 3 * 9 = 3^3.

%e 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

%p f:= proc(n) local F,E;

%p    F:= ifactors(n)[2];

%p    E:= F[..,2];

%p    if max(E) >= 3 then return NULL fi;

%p    mul(F[i,1]^(3-E[i]),i=1..nops(F));

%p end proc:

%p map(f, [$1..1000]); # _Robert Israel_, Nov 09 2016

%t 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 *)

%t 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 *)

%o (PARI) \\ A list of about n terms (a little more probably).

%o 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

%o (Python)

%o from math import prod

%o from sympy import mobius, factorint, integer_nthroot

%o def A277802(n):

%o     def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))

%o     m, k = n, f(n)

%o     while m != k:

%o         m, k = k, f(k)

%o     return prod(p**(-e%3) for p, e in factorint(m).items()) # _Chai Wah Wu_, Aug 05 2024

%Y Cf. A002117, A048798, A004709.

%K nonn,look

%O 1,2

%A _Peter Kagey_, Oct 31 2016