A320965 -id:A320965 - cp's OEIS frontend

A320966 Powerful numbers A001694 divisible by a cube > 1.

8, 16, 27, 32, 64, 72, 81, 108, 125, 128, 144, 200, 216, 243, 256, 288, 324, 343, 392, 400, 432, 500, 512, 576, 625, 648, 675, 729, 784, 800, 864, 968, 972, 1000, 1024, 1125, 1152, 1296, 1323, 1331, 1352, 1372, 1568, 1600, 1728, 1800, 1936, 1944, 2000, 2025, 2048, 2187, 2197, 2304, 2312, 2401, 2500

Offset: 1

Hugo Pfoertner, Oct 25 2018

nonn

Powerful numbers that are not squares of squarefree numbers. - Amiram Eldar, Jun 25 2022

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000578, A320965.
Intersection of A001694 and A046099.
A001694 \ A062503.

Select[Range[2500], (m = MinMax[FactorInteger[#][[;; , 2]]])[[1]] > 1 && m[[2]] > 2 &] (* Amiram Eldar, Jun 25 2022 *)

isA001694(n)=n=factor(n)[, 2]; for(i=1, #n, if(n[i]==1, return(0))); 1 \\ from Charles R Greathouse IV
isA046099(n)=n=factor(n)[, 2]; for(i=1, #n, if(n[i]>2, return(1)));0
for (k=1,2500,if(isA001694(k)&&isA046099(k),print1(k,", ")))

from math import isqrt
from sympy import mobius, integer_nthroot
def A320966(n):
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x):
        c, l = n+x+squarefreepi(isqrt(x))-squarefreepi(integer_nthroot(x,3)[0]), 0
        j = isqrt(x)
        while j>1:
            k2 = integer_nthroot(x//j**2,3)[0]+1
            w = squarefreepi(k2-1)
            c -= j*(w-l)
            l, j = w, isqrt(x//k2**3)
        return c+l
    return bisection(f,n,n) # Chai Wah Wu, Sep 15 2024

Sum_{n>=1} 1/a(n) = zeta(2)*zeta(3)/zeta(6) - 15/Pi^2 = 0.4237786821... . - Amiram Eldar, Jun 25 2022

A062320 Nonsquarefree numbers squared. A013929(n)^2.

Original entry on oeis.org

16, 64, 81, 144, 256, 324, 400, 576, 625, 729, 784, 1024, 1296, 1600, 1936, 2025, 2304, 2401, 2500, 2704, 2916, 3136, 3600, 3969, 4096, 4624, 5184, 5625, 5776, 6400, 6561, 7056, 7744, 8100, 8464, 9216, 9604, 9801, 10000, 10816, 11664, 12544, 13456

Offset: 1

Jason Earls, Jul 05 2001

A008966(A037213(a(n))) = 0. - Reinhard Zumkeller, Sep 03 2015

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A013929, A008966, A037213, A320965.

Squares in A046099.

a062320 = (^ 2) . a013929 -- Reinhard Zumkeller, Sep 03 2015

for(n=1,55, if(issquarefree(n), n+1,print(n^2)))

n=-1; for (m=1, 10^9, if (!issquarefree(m), write("b062320.txt", n++, " ", m^2); if (n==1000, break))) \\ Harry J. Smith, Aug 04 2009

is(n)=issquare(n,&n) && !issquarefree(n) \\ Charles R Greathouse IV, Sep 18 2015

from math import isqrt
from sympy import mobius
def A062320(n):
    def f(x): return n+1+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f)**2 # Chai Wah Wu, Aug 31 2024

Sum_{n>=1} 1/a(n) = Pi^2/6 - 15/Pi^2. - Amiram Eldar, Jul 16 2020

More terms from Larry Reeves (larryr(AT)acm.org), Jul 11 2001

Offset corrected by Andrew Howroyd, Sep 18 2024

A321070 Squares divisible by more than one cube > 1.

Original entry on oeis.org

64, 256, 576, 729, 1024, 1296, 1600, 2304, 2916, 3136, 4096, 5184, 6400, 6561, 7744, 9216, 10000, 10816, 11664, 12544, 14400, 15625, 16384, 18225, 18496, 20736, 23104, 25600, 26244, 28224, 30976, 32400, 33856, 35721, 36864, 38416, 40000, 43264, 46656, 50176

Offset: 1

Hugo Pfoertner, Oct 27 2018

nonn

			a(1) = 64 because 16^2 is divisible by 2^3 and by 4^3.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A000290, A000578, A062503, A062320, A216427, A320965.

Select[Range[225]^2, Max[(e = FactorInteger[#][[;; , 2]])] > 4 || (Length[e] > 1 && Sort[e, Greater][[2]] > 2) &] (* Amiram Eldar, Jun 25 2022 *)

iscubes(n) = {my(nb = 0); fordiv(n, d, if ((d>1) && ispower(d, 3), nb++; if (nb > 1, return(1))););}
isok(n) = issquare(n) && iscubes(n); \\ Michel Marcus, Oct 27 2018

From Amiram Eldar, Jun 25 2022: (Start)
Equals A000290 \ (union of A062503 and A320965).
Sum_{n>=1} 1/a(n) = Pi^2/6 - (15/Pi^2) * (1 + Sum_{k>=2} (-1)^k * P(2*k)) = 0.029082273527998239268... . (End)

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A320966 Powerful numbers A001694 divisible by a cube > 1.

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A062320 Nonsquarefree numbers squared. A013929(n)^2.

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A321070 Squares divisible by more than one cube > 1.

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