A322449 -id:A322449 - cp's OEIS frontend

A117453 Perfect powers in more than one way.

1, 16, 64, 81, 256, 512, 625, 729, 1024, 1296, 2401, 4096, 6561, 10000, 14641, 15625, 16384, 19683, 20736, 28561, 32768, 38416, 46656, 50625, 59049, 65536, 83521, 104976, 117649, 130321, 160000, 194481, 234256, 262144, 279841, 331776, 390625

Offset: 1

Eric W. Weisstein, Mar 16 2006

nonn

Corresponding values of ways for a(n) in A175066(n) for n >= 2. - Jaroslav Krizek, Jan 23 2010

Perfect powers expressible as m^k with k composite. - Charlie Neder, Mar 02 2019

			16 = 2^4 = 4^2.

Donovan Johnson, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Perfect Power

Cf. A001597, A322449.

s = Split@ Sort@ Flatten@ Table[ n^i, {n, 2, Sqrt@456975}, {i, 2, Log[n, 456975]}]; Union@ Flatten@ Select[s, Length@ # > 1 &] (* Robert G. Wilson v, Apr 12 2006 *)

from sympy import mobius, integer_nthroot, primerange
def A117453(n):
    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): return int(n+sum(mobius(k)*(integer_nthroot(x,k)[0]-1+sum(integer_nthroot(x,p*k)[0]-1 for p in primerange((x//k).bit_length()))) for k in range(1,x.bit_length())))
    return bisection(f,n,n) # Chai Wah Wu, Nov 24 2024

More terms from Robert G. Wilson v, Apr 12 2006

A322448 Numbers whose prime factorization contains at least one composite exponent.

Original entry on oeis.org

16, 48, 64, 80, 81, 112, 144, 162, 176, 192, 208, 240, 256, 272, 304, 320, 324, 336, 368, 400, 405, 432, 448, 464, 496, 512, 528, 560, 567, 576, 592, 624, 625, 648, 656, 688, 704, 720, 729, 752, 768, 784, 810, 816, 832, 848, 880, 891, 912, 944, 960, 976, 1008

Offset: 1

Alois P. Heinz, Dec 08 2018

nonn

The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^4 + Sum_{q prime >= 5} 1/p^q - 1/p^(q-1)) = 0.05328066264472198953... (using the method of Shevelev, 2016). - Amiram Eldar, Nov 08 2020

			16 = 2^4 is a term because 4 is a composite exponent here.

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Vladimir Shevelev, A fast computation of density of exponentially S-numbers, arXiv:1602.04244 [math.NT], 2016.

Cf. A002808, A322449.

a:= proc(n) option remember; local k; for k from 1+
      `if`(n=1, 0, a(n-1)) while andmap(i-> i[2]=1 or
       isprime(i[2]), ifactors(k)[2]) do od; k
    end:
seq(a(n), n=1..70);

Select[Range[1000], AnyTrue[FactorInteger[#][[;; , 2]], CompositeQ] &] (* Amiram Eldar, Nov 08 2020 *)

isok(m) = #select(x->((x>1) && !isprime(x)), factor(m)[,2]) > 0; \\ Michel Marcus, Dec 02 2020

A374588 Numbers whose maximum exponent in their prime factorization is a composite number.

Original entry on oeis.org

Offset: 1

Amiram Eldar, Jul 12 2024

Subsequence of A322448 and first differs from it at n = 138: A322448(138) = 2592 = 2^5 * 3^4 is not a term of this sequence.

The asymptotic density of this sequence is d = Sum_{k composite} (1/zeta(k+1) - 1/zeta(k)) = 0.05296279266796920306... . The asymptotic density of this sequence within the nonsquarefree numbers (A013929) is d / (1 - 1/zeta(2)) = 0.13508404411123191108... .

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

Cf. A002808, A013661, A051903, A229099.
Complement of A074661 within A013929.
Subsequence of A322448 and A322449 \ {1}.
Similar sequences: A368714, A369937, A369938, A369939, A374589, A374590.

filter:= proc(n) local m;
  m:= max(ifactors(n)[2][..,2]);
  m > 1 and not isprime(m)
end proc:
select(filter, [$1..10000]); # Robert Israel, Jul 14 2024

Select[Range[1200], CompositeQ[Max[FactorInteger[#][[;; , 2]]]] &]

iscomposite(n) = n > 1 && !isprime(n);
is(n) = n > 1 && iscomposite(vecmax(factor(n)[, 2]));

A374291 Squares of powerful numbers.

Original entry on oeis.org

1, 16, 64, 81, 256, 625, 729, 1024, 1296, 2401, 4096, 5184, 6561, 10000, 11664, 14641, 15625, 16384, 20736, 28561, 38416, 40000, 46656, 50625, 59049, 65536, 82944, 83521, 104976, 117649, 130321, 153664, 160000, 186624, 194481, 234256, 250000, 262144, 279841, 331776

Offset: 1

Amiram Eldar, Jul 02 2024

First differs from A340588 at n = 12.
4-full (or 3-full) squares.
Numbers whose exponents in their prime factorization are all even numbers >= 4.
This sequence is closed under multiplication.
The sequence {A000290(n)*A078615(A000290(n)), n>=1} is a permutation of this sequence, and the sequence {a(n)/A078615(a(n)), n>=1} is a permutation of {A000290(n), n>=1}.
The sequence {A335988(n)*A007947(A335988(n)), n>=1} is a permutation of this sequence, and the sequence {a(n)/A007947(a(n)), n>=1} is a permutation of A335988.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Index entries for sequences related to powerful numbers.

Intersection of A000290 and A036967 (or A036966).
Intersection of A000290 and A337050.
Subsequence of A322449.
Cf. A000290, A001694, A007947, A078615, A335988, A340588.

powQ[n_] := n==1 || AllTrue[FactorInteger[n][[;; , 2]], # > 1 &]; Select[Range[600], powQ]^2

is(k) = issquare(k) && ispowerful(sqrtint(k));

from math import isqrt
from sympy import mobius, integer_nthroot
def A374291(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, 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)
        c -= squarefreepi(integer_nthroot(x, 3)[0])-l
        return c
    return bisection(f,n,n)**2 # Chai Wah Wu, Sep 10 2024

a(n) = A000290(A001694(n)) = A001694(n)^2.
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2*(p^2-1))) = zeta(4)*zeta(6)/zeta(12) = 15015/(1382*Pi^2) = 1.10082313486953808844... .
Sum_{n>=1} 1/a(n)^s =  Product_{p prime} (1 + 1/(p^(2*s)*(p^(2*s)-1))) = zeta(4*s)*zeta(6*s)/zeta(12*s), for s > 1/4.

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A117453 Perfect powers in more than one way.

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A322448 Numbers whose prime factorization contains at least one composite exponent.

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A374588 Numbers whose maximum exponent in their prime factorization is a composite number.

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A374291 Squares of powerful numbers.

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