A036967 -id:A036967 - cp's OEIS frontend

A374291 Squares of powerful numbers.

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.

A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 1, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 1, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 3, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 1, 65, 66, 67, 68, 69

Offset: 1

Amiram Eldar, Jun 15 2025

First differs from A053165 at n = 32 = 2^5: a(32) = 1 while A053165(32) = 2.
First differs from A383764 at n = 32 = 2^5: a(32) = 1 while A383764(32) = 32.
Equivalently, a(n) is the least divisor d of n such that n/d is a 4-full number (A036967).

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

Cf. A013661, A036967, A046100, A053165, A058035, A077610, A383764.

The largest unitary divisor of n that is: A000265 (odd), A006519 (power of 2), A055231 (squarefree), A057521 (powerful), A065330 (5-rough), A065331 (3-smooth), A350388 (square), A350389 (exponentially odd), A360539 (cubefree), A360540 (cubefull), A366126 (cube), A367168 (exponentially 2^n), this sequence (biquadratefree).

f[p_, e_] := If[e < 4, p^e, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i=1, #f~, if(f[i, 2] < 4, f[i, 1]^f[i, 2], 1)); }

a(n) = 1 if and only if n is a 4-full number (A036967).
a(n) = n if and only if n is a biquadratefree number (A046100).
Multiplicative with a(p^e) = p^e if e <= 3, and 1 otherwise.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(-s) + p^(2-2*s) - p^(1-2*s) - p^(2-3*s) + p^(3-3*s) - p^(3-4*s) + p^(-4*s)).
Sum_{k=1..n} a(k) ~ c * zeta(2) * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^6 + 1/p^8 - 1/p^9) = 0.56331392082909224894... .

A360844 a(n) is the least k-full number that is sandwiched between twin primes.

Original entry on oeis.org

4, 432, 2592, 139968, 139968, 174960000000, 56358560858112, 84537841287168, 578415690713088, 578415690713088, 1141260857376768, 61628086298345472, 61628086298345472, 61628086298345472, 322850407500000000000000000000, 322850407500000000000000000000, 62518864539857068333550694039552

Offset: 2

Amiram Eldar, Feb 23 2023

nonn

k-full number is a number m such that if a prime p divides m then so does p^k. All the exponents in the canonical prime factorization of a k-full number are not smaller than k.
a(2)-a(15) are the terms below 3*10^19. Except for a(7) = 174960000000, they are all 3-smooth numbers (A003586, and thus they are terms of A027856). Are there other terms that are not 3-smooth?
a(168) = 2^176 * 3^173 * 7^168 is the first term that is not 5-smooth. - Bert Dobbelaere, Feb 24 2023

			The first 3 terms, their factorizations and the corresponding twin primes are:
  n |   a(n)  prime factorization  A051904(a(n))  {a(n)-1, a(n)+1}
  ----------------------------------------------------------------
  2 |     4                  2^2              2             {3, 5}
  3 |   432            2^4 * 3^3              3         {431, 433}
  4 |  2592            2^5 * 3^4              4       {2591, 2593}

Bert Dobbelaere, Table of n, a(n) for n = 2..200
Eric Weisstein's World of Mathematics, Twin Primes.
Wikipedia, Powerful number: Generalization.
Index entries for sequences related to powerful numbers.

Cf. A001097, A001694, A014574, A036966, A036967, A069492, A069493.
Cf. A113839, A360840, A360841, A360842, A360843.
Cf. A027856, A003586, A051904.

More terms from Bert Dobbelaere, Feb 24 2023

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A374291 Squares of powerful numbers.

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A385007 The largest unitary divisor of n that is a biquadratefree number (A046100).

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A360844 a(n) is the least k-full number that is sandwiched between twin primes.

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