A153159 -id:A153159 - 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

A153147 a(n) = A007916(n)^3.

Original entry on oeis.org

8, 27, 125, 216, 343, 1000, 1331, 1728, 2197, 2744, 3375, 4913, 5832, 6859, 8000, 9261, 10648, 12167, 13824, 17576, 21952, 24389, 27000, 29791, 35937, 39304, 42875, 50653, 54872, 59319, 64000, 68921, 74088, 79507, 85184, 91125, 97336, 103823

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 19 2008

nonn

2^3=8, 3^3=27, 4^3=64=2^6 is not in the sequence, 5^3=125, 6^3=216, ...

Cf. A000578, A007916, A153157, A153158, A153159, A153160, A062838.

Select[Range[2,100],GCD@@Last/@FactorInteger@#==1&]^3

from sympy import mobius, integer_nthroot
def A153147(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**3

Edited by Ray Chandler, Dec 22 2008

A153157 a(n) = A007916(n)^4.

Original entry on oeis.org

16, 81, 625, 1296, 2401, 10000, 14641, 20736, 28561, 38416, 50625, 83521, 104976, 130321, 160000, 194481, 234256, 279841, 331776, 456976, 614656, 707281, 810000, 923521, 1185921, 1336336, 1500625, 1874161, 2085136, 2313441, 2560000, 2825761

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 19 2008

nonn

2^4=16,3^4=81,4^4=256=2^8 is not in the sequence,5^4=625,6^4=1296,...

Cf. A007916, A153147, A153158, A153159, A153160, A113849.

Select[Range[2,100],GCD@@Last/@FactorInteger@#==1&]^4

from sympy import mobius, integer_nthroot
def A153157(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**4 # Chai Wah Wu, Nov 21 2024

Edited by Ray Chandler, Dec 22 2008

A153160 a(n) = A007916(n)^6.

Original entry on oeis.org

64, 729, 15625, 46656, 117649, 1000000, 1771561, 2985984, 4826809, 7529536, 11390625, 24137569, 34012224, 47045881, 64000000, 85766121, 113379904, 148035889, 191102976, 308915776, 481890304, 594823321, 729000000, 887503681

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 19 2008

nonn

Cf. A007916, A153147, A153157, A153158, A153159, A113851.

Select[Range[2,100],GCD@@Last/@FactorInteger@#==1&]^6

from sympy import mobius, integer_nthroot
def A153160(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**6 # Chai Wah Wu, Nov 21 2024

Edited and extended by Ray Chandler, Dec 22 2008

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

Python

Formula

Extensions

A153147 a(n) = A007916(n)^3.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

Extensions

A153157 a(n) = A007916(n)^4.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Python

Extensions

A153160 a(n) = A007916(n)^6.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Python

Extensions