A072777 -id:A072777 - cp's OEIS frontend

A384517 Nonsquarefree numbers that are squarefree numbers raised to an even power.

4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 169, 196, 225, 256, 289, 361, 441, 484, 529, 625, 676, 729, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1681, 1764, 1849, 2116, 2209, 2401, 2601, 2809, 3025, 3249, 3364, 3481, 3721, 3844, 4096, 4225, 4356

Offset: 1

Amiram Eldar, Jun 01 2025

nonn

Differs from its subsequence A340674 by having the terms 64, 729, 1024, 4096, .... .

Numbers whose prime factorization exponents are equal and even.

Intersection of A000290 and A072777.
Equals A072777 \ A384518.
A340674 is a subsequence.
Cf. A005117, A062770, A072774, A231273, A231327.

Select[Range[2, 100], SameQ @@ FactorInteger[#][[;;, 2]] &]^2

isok(k) = {my(s, e = ispower(k, , &s)); !(e % 2) && issquarefree(s);}

from math import isqrt
from sympy import mobius, integer_nthroot
def A384517(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1		
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return n+x-sum(g(integer_nthroot(x,e)[0])-1 for e in range(2,x.bit_length(),2))
    return bisection(f,n,n) # Chai Wah Wu, Jun 01 2025

a(n) = A062770(n)^2 = A072774(n+1)^2.

Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2*k)/zeta(4*k)-1) = Sum{k>=1} (A231327(k)/(A231273(k)*Pi^(2*k)) - 1) = 0.62022193512079649421... .

A384518 Nonsquarefree numbers that are squarefree numbers raised to an odd power.

Original entry on oeis.org

8, 27, 32, 125, 128, 216, 243, 343, 512, 1000, 1331, 2048, 2187, 2197, 2744, 3125, 3375, 4913, 6859, 7776, 8192, 9261, 10648, 12167, 16807, 17576, 19683, 24389, 27000, 29791, 32768, 35937, 39304, 42875, 50653, 54872, 59319, 68921, 74088, 78125, 79507, 97336, 100000

Offset: 1

Amiram Eldar, Jun 01 2025

nonn

Subsequence of A097054 and first differs from it at n = 12: A097054(12) = 1728 = 2^6 * 3^3 is not a term of this sequence.

Numbers whose prime factorization exponents are equal, odd and larger than 1.

Intersection of A072777 and A268335.
Equals A072777 \ A384517.
Subsequence of A097054.
Cf. A005117.

Select[Range[10^5], Length[(u = Union[FactorInteger[#][[;; , 2]]])] == 1 && u[[1]] > 1 && OddQ[u[[1]]] &]

isok(k) = {my(s, e = ispower(k, , &s)); e % 2 && issquarefree(s);}

from math import isqrt
from sympy import mobius, integer_nthroot
def A384518(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1		
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def g(x): return sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def f(x): return n+x-sum(g(integer_nthroot(x,e)[0])-1 for e in range(3,x.bit_length(),2))
    return bisection(f,n,n) # Chai Wah Wu, Jun 01 2025

Sum_{n>=1} 1/a(n) = Sum_{k>=1} (zeta(2*k+1)/zeta(4*k+2)-1) = 0.22841193284408713846... .

A076292 Perfect powers with squarefree root.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 169, 196, 216, 225, 243, 256, 289, 343, 361, 441, 484, 512, 529, 625, 676, 729, 841, 900, 961, 1000, 1024, 1089, 1156, 1225, 1296, 1331, 1369, 1444, 1521, 1681, 1764, 1849, 2048, 2116, 2187

Offset: 1

Reinhard Zumkeller, Nov 06 2002

nonn

			A001597(17) = 144 = (3*2^2)^2 is not a term.

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

Cf. A001597, A025478, A005117, A007947, A072777.

q[n_] := n == 1 || Length[(u = Union[FactorInteger[n][[;;,2]]])] == 1 && u[[1]] > 1; Select[Range[2000], q] (* Amiram Eldar, Jan 01 2022 *)

is(n)=n==1 || (ispower(n,,&n) && issquarefree(n)) \\ Charles R Greathouse IV, Oct 16 2015

A025478(a(n)) = A007947(a(n)).

A375934 Numbers whose prime factorization has a second-largest exponent that equals 1.

Original entry on oeis.org

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 180, 184, 188, 189, 192, 198, 204

Offset: 1

Amiram Eldar, Sep 03 2024

First differs from A332785 at n = 112: A332785(112) = 360 = 2^3 * 3^2 * 5 is not a term of this sequence.
First differs from A317616 at n = 38: A317616(38) = 144 = 2*4 * 3^2 is not a term of this sequence.
Numbers k such that A375933(k) = 1.
Numbers of the form s1 * s2^e, where s1 and s2 are coprime squarefree numbers that are both larger than 1, and e >= 2.
The asymptotic density of this sequence is Sum_{e>=2} d(e) = 0.36113984820338109927..., where d(e) = Product_{p prime} (1 - 1/p^2 + 1/p^e - 1/p^(e+1)) - Product_{p prime} (1 - 1/p^(e+1)) is the asymptotic density of terms k with A051903(k) = e >= 2.

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

Equals A375142 \ A072777.
Subsequence of A013929.
Subsequences: A054753, A065036, A072357, A095990, A096156, A178739, A178740, A179664, A179668, A179692, A189987, A360793, A375076.
Cf. A001221, A005117, A051903, A051904, A072777, A317616, A332785, A375142, A375933.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[0, Max[Select[e, # < Max[e] &]]] == 1]; Select[Range[300], q]

is(n) = if(n == 1, 0, my(e = factor(n)[,2]); e = select(x -> x < vecmax(e), e); if(#e == 0, 0, vecmax(e) == 1));

A051904(a(n)) = 1.
A051903(a(n)) >= 2.
A001221(a(n)) = 2.

A375143 Numbers whose prime factorization has a minimum exponent that is larger than 1 and is 1 less than the maximum exponent.

Original entry on oeis.org

72, 108, 200, 392, 432, 500, 648, 675, 968, 1125, 1323, 1352, 1372, 1800, 2000, 2312, 2592, 2700, 2888, 3087, 3267, 3528, 3888, 4232, 4500, 4563, 5000, 5292, 5324, 5400, 5488, 6125, 6728, 7688, 7803, 8575, 8712, 8788, 9000, 9747, 9800, 10125, 10584, 10952, 11979

Offset: 1

Amiram Eldar, Aug 01 2024

Numbers k such that 2 <= A051904(k) = A051903(k) - 1.

Numbers that are product of two coprime nonsquarefree powers of squarefree numbers (A072777) with consecutive exponents.

			72 = 2^3 * 3^2 is a term since A051904(72) = 2 is larger than 1 and is 1 less than A051903(72) = 3.

Cf. A051903, A051904, A072777.
Subsequence of A001694.
Subsequences: A143610, A167747 \ {1, 2, 12}, A093136 \ {1, 2, 20}, A179666, A179702, A190472, A375073.

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, 2 <= Min[e] == Max[e] - 1]; Select[Range[12000], q]

is(k) = {my(e = factor(k)[,2]); k > 1 && 2 <= vecmin(e) && vecmin(e) + 1 == vecmax(e);}

Sum_{n>=1} 1/a(n) = Sum_{k>=2} f(k) = 0.053695635500385312854..., where f(k) = Product_{p prime} (1 + 1/p^k + 1/p^(k+1)) - zeta(k)/zeta(2*k) - zeta(k+1)/zeta(2*k+2) + 1 is the sum of reciprocals of the subset of numbers m with A051904(m) = k.

cp's OEIS Frontend

A384517 Nonsquarefree numbers that are squarefree numbers raised to an even power.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A384518 Nonsquarefree numbers that are squarefree numbers raised to an odd power.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A076292 Perfect powers with squarefree root.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A375934 Numbers whose prime factorization has a second-largest exponent that equals 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A375143 Numbers whose prime factorization has a minimum exponent that is larger than 1 and is 1 less than the maximum exponent.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula