A383004 -id:A383004 - cp's OEIS frontend

A383005 Exponent of the highest power of 2 dividing the n-th biquadratefree number.

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0

Offset: 1

Amiram Eldar, Apr 12 2025

First differs from A254990 at n = 31.

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

Cf. A046100, A007814, A254990, A373550, A383004.

biqFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 4 &]; IntegerExponent[Select[Range[200], biqFreeQ], 2]

isbiqfree(n) = {my(f = factor(n)); for(i=1, #f~, if(f[i, 2] > 3, return (0))); 1; }
list(lim) = for(k = 1, lim, if(isbiqfree(k), print1(valuation(k, 2), ", ")));

a(n) = A007814(A046100(n)).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 11/15.
In general, the asymptotic mean of the exponent of the highest power of 2 dividing the n-th k-free number (number that is not divisible by a k-th power other than 1), for k >= 2, is 1 - k/(2^k-1).

A383009 Indices of the even terms in the sequence of cubefree numbers.

Original entry on oeis.org

2, 4, 6, 9, 11, 13, 16, 18, 20, 23, 24, 26, 29, 31, 33, 36, 38, 40, 43, 45, 49, 51, 53, 56, 58, 60, 63, 65, 67, 69, 71, 73, 76, 78, 80, 83, 85, 87, 90, 93, 96, 98, 100, 103, 105, 106, 109, 111, 113, 115, 117, 119, 122, 124, 126, 129, 131, 133, 137, 139, 142, 144

Offset: 1

Amiram Eldar, Apr 12 2025

The asymptotic density of this sequence is 3/7.

In general, the asymptotic density of the indices of the even terms in the sequence of k-free numbers (numbers that are not divisible by a k-th power other than 1), for k >= 2, is (2^(k-1)-1)/(2^k-1).

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

Cf. A004709, A381822, A383004, A383008.

cubeFreeQ[n_] := AllTrue[FactorInteger[n][[;; , 2]], # < 3 &]; Position[Select[Range[250], cubeFreeQ], _?EvenQ] // Flatten

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

A383004(a(n)) > 0.

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... .

cp's OEIS Frontend

A383005 Exponent of the highest power of 2 dividing the n-th biquadratefree number.

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A383009 Indices of the even terms in the sequence of cubefree numbers.

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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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