A383005 -id:A383005 - cp's OEIS frontend

A383004 Exponent of the highest power of 2 dividing the n-th cubefree number.

0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1

Offset: 1

Amiram Eldar, Apr 12 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A004709, A007814, A373550, A383005, A383009.

cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; IntegerExponent[Select[Range[200], cubeFreeQ], 2]

iscubefree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 2, return (0))); 1; }
list(lim) = for(k = 1, lim, if(iscubefree(k), print1(valuation(k, 2), ", ")));

a(n) = A007814(A004709(n)).
A383009(a(n)) > 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4/7.

A386536 Exponent of the highest power of 2 dividing the n-th number that is cubefree but not squarefree.

Original entry on oeis.org

2, 0, 2, 1, 2, 0, 2, 2, 2, 0, 0, 1, 2, 2, 0, 2, 0, 2, 2, 1, 2, 1, 0, 2, 2, 0, 0, 2, 1, 2, 2, 0, 2, 1, 0, 2, 2, 0, 0, 2, 0, 2, 2, 2, 1, 2, 0, 2, 2, 0, 2, 1, 2, 1, 2, 0, 2, 2, 0, 2, 0, 2, 0, 2, 0, 2, 1, 2, 1, 2, 0, 2, 0, 2, 0, 1, 2, 1, 2, 1, 2, 0, 0, 2, 0, 2, 2

Offset: 1

Amiram Eldar, Jul 25 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002117, A013661, A007814, A067259.

Similar sequences: A383004, A383005, A383006, A383007, A383029, A383032, A386537.

IntegerExponent[Select[Range[400], Max[FactorInteger[#][[;; , 2]]] == 2 &], 2]

list(lim) = for(k = 1, lim, if(k > 1 && vecmax(factor(k)[,2]) == 2, print1(valuation(k, 2), ", ")));

from math import isqrt
from sympy import integer_nthroot, mobius
def A386536(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while f(kmin) < kmin: kmin >>= 1		
        kmin = max(kmin,kmax >> 1)
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x+sum(mobius(k)*(x//k**2-x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))+sum(mobius(k)*(x//k**2) for k in range(integer_nthroot(x,3)[0]+1,isqrt(x)+1)))
    return ((m:=bisection(f,n,n))-1&~m).bit_length() # Chai Wah Wu, Jul 25 2025

a(n) = A007814(A067259(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (4/(7*zeta(3)) - 1/(3*zeta(2)))/(1/zeta(3) - 1/zeta(2)) = 1.2176665... .

A386537 Exponent of the highest power of 2 dividing the n-th number whose prime factorization exponents are all powers of 2 (A138302).

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 0, 1, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 1

Offset: 1

Amiram Eldar, Jul 25 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A007404, A007814, A138302.

Similar sequences: A383004, A383005, A383006, A383007, A383029, A383032, A386536.

exp2nQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # == 2^IntegerExponent[#, 2] &];
IntegerExponent[Select[Range[200], exp2nQ], 2]

isexp2n(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] >> valuation(f[i, 2], 2) > 1, return (0))); 1;}
list(lim) = for(k = 1, lim, if(isexp2n(k), print1(valuation(k, 2), ", ")));

a(n) = A007814(A138302(n)).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = (1 + Sum_{k>=0} (2^k + 1)/2^(2^k)) / (1 + Sum_{k>=0} 1/2^(2^k)) - 1 = 0.70550483007968767769... .

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A383004 Exponent of the highest power of 2 dividing the n-th cubefree number.

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A386536 Exponent of the highest power of 2 dividing the n-th number that is cubefree but not squarefree.

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A386537 Exponent of the highest power of 2 dividing the n-th number whose prime factorization exponents are all powers of 2 (A138302).

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