A231273 -id:A231273 - 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... .

A231327 Denominator of rational component of zeta(4n)/zeta(2n).

Original entry on oeis.org

1, 15, 105, 675675, 34459425, 16368226875, 218517792968475, 30951416768146875, 694097901592400930625, 23383376494609715287281703125, 2289686345687357378035370971875, 219012470258383844016431785453125, 4791965046290912124048163518904807546875

Offset: 0

Leo Depuydt, Nov 07 2013

Denominator of a close variant of Euler's infinite prime product zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)-1), namely with all minus signs changed into plus signs, as follows: zeta(4n)/zeta(2n) = Product_{k>=1} prime(k)^(2n)/(prime(k)^(2n)+1).
For a detailed account of the results in question, including proof and relation to the zeta function, see the PDF file submitted as supporting material in A231273.
The reference to Apostol below is a discussion of the equivalence of 1) zeta(2s)/zeta(s) and 2) a related infinite prime product, that is, Product_{sigma>1} prime(n)^s/(prime(n)^s + 1), with s being a complex variable such that s = sigma + i*t where sigma and t are real (following Riemann), using a type of proof different from the one posted below involving zeta(4n)/zeta(2n). - Leo Depuydt, Nov 22 2013
Denominator of B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!) where B(n) are the Bernoulli numbers (see A027641 and A027642). - Robert Israel, Aug 22 2014

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976, p. 231.

Cf. A231273 (the corresponding numerator).
Cf. A114362 and A114363 (closely related results).
Cf. A001067, A046968, A046988, A098087, A141590, and A156036 (same number sequence as found in numerator, though in various transformations (alternation of sign, intervening numbers, and so on)).
Cf. A027641 and A027642.

seq(denom(bernoulli(4*n)*4^n*(2*n)!/(bernoulli(2*n)*(4*n)!)),n=0..100); # Robert Israel, Aug 22 2014

Denominator[Table[Zeta[4 n]/Zeta[2 n], {n, 0, 15}]] (* T. D. Noe, Nov 15 2013 *)

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

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