A378168 -id:A378168 - cp's OEIS frontend

A153158 a(n) = A007916(n)^2.

4, 9, 25, 36, 49, 100, 121, 144, 169, 196, 225, 289, 324, 361, 400, 441, 484, 529, 576, 676, 784, 841, 900, 961, 1089, 1156, 1225, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2500, 2601, 2704, 2809, 2916, 3025, 3136, 3249, 3364, 3481

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 19 2008

nonn

A378168(n) is the number of terms <= 10^n. - Chai Wah Wu, Nov 21 2024

			2^2 = 4, 3^2 = 9, 4^2 = 16 = 2^4 is not in the sequence, 5^2 = 25, 6^2 = 36, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A007916, A153147, A153157, A153159, A153160, A062503, A076467, A378168, A378287.

a153158 n = a153158_list !! (n-1)
a153158_list = filter ((== 2) . foldl1 gcd . a124010_row) [2..]
-- Reinhard Zumkeller, Apr 13 2012

q:= n-> is(igcd(seq(i[2], i=ifactors(n)[2]))=2):
select(q, [i^2$i=2..60])[];  # Alois P. Heinz, Nov 26 2024

Select[Range[2,100],GCD@@Last/@FactorInteger@#==1&]^2

from sympy import mobius, integer_nthroot
def A153158(n):
    def f(x): return int(n+1-sum(mobius(k)*(integer_nthroot(x,k)[0]-1) for k in range(2,x.bit_length())))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m**2 # Chai Wah Wu, Aug 13 2024

GCD(exponents in prime factorization of a(n)) = 2, cf. A124010. - Reinhard Zumkeller, Apr 13 2012
Sum_{n>=1} 1/a(n) = zeta(2) - 1 - Sum_{k>=2} mu(k)*(1 - zeta(2*k)) = 0.5444587396... - Amiram Eldar, Jul 02 2022
Intersection of A000290 and A378287. Squares that are not of the form m^k for some k>=3. - Chai Wah Wu, Nov 21 2024

Edited by Ray Chandler, Dec 22 2008

A377934 a(n) is the number of perfect powers m^k with k>=3 (A076467) <= 10^n.

Original entry on oeis.org

1, 2, 7, 17, 38, 75, 152, 306, 616, 1260, 2598, 5401, 11307, 23798, 50316, 106776, 227236, 484737, 1036002, 2217529, 4752349, 10194727, 21887147, 47020054, 101065880, 217325603, 467484989, 1005881993, 2164843035, 4660016778, 10032642455, 21602193212, 46518438071

Offset: 0

Hugo Pfoertner, Nov 24 2024

nonn

			a(0) = 1: 1^k with any k>2 (<= 10^0);
a(1) = 2: 1 and 2^3 (<=10^1);
a(2) = 7: 2 powers <= 10 and 16, 27, 32, 64, 81 (<=10^2).

Chai Wah Wu, Table of n, a(n) for n = 0..2999

Cf. A070428, A076467, A378168.

from math import gcd
from sympy import integer_nthroot, mobius
def A377934(n): return int(integer_nthroot(10**(n//(a:=gcd(n,4))),4//a)[0]-sum(mobius(k)*(integer_nthroot(10**(n//(b:=gcd(n,k))),k//b)[0]+integer_nthroot(10**(n//(c:=gcd(n,d:=k<<1))),d//c)[0]-2) for k in range(3,(10**n).bit_length()))) # Chai Wah Wu, Nov 24 2024

a(n) = 10^n - Sum_{k=1..floor(log2(10^n))} mu(k)*(floor(10^(n/k))+floor(10^(n/(2k)))-2). - Chai Wah Wu, Nov 24 2024

a(28) onwards from Chai Wah Wu, Nov 24 2024

cp's OEIS Frontend

A153158 a(n) = A007916(n)^2.

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