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A380857
Squares of numbers that are neither squarefree nor prime powers.
1
144, 324, 400, 576, 784, 1296, 1600, 1936, 2025, 2304, 2500, 2704, 2916, 3136, 3600, 3969, 4624, 5184, 5625, 5776, 6400, 7056, 7744, 8100, 8464, 9216, 9604, 9801, 10000, 10816, 11664, 12544, 13456, 13689, 14400, 15376, 15876, 17424, 18225, 18496, 19600, 20736
OFFSET
1,1
COMMENTS
Proper subset of A359280 which is a proper subset of A286708 (powerful numbers that are not prime powers, a proper subset of A126706).
Does not intersect A362605.
LINKS
FORMULA
a(n) = A126706(n)^2.
Sum_{n>=1} 1/a(n) = Pi^2/6 - 15/Pi^2 - Sum_{p prime} 1/(p^2*(p^2-1)) = A013661 - A082020 + A085548 - A154945 = 0.025670434597226178881... . - Amiram Eldar, Feb 08 2025
MATHEMATICA
Select[Range[150], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]^2
PROG
(PARI) isok(k) = !issquarefree(k) && !isprimepower(k); \\ A126706
apply(sqr, select(isok, [1..200])) \\ Michel Marcus, Feb 07 2025
(Python)
from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A380857(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 f(x): return int(n+sum(primepi(integer_nthroot(x, k)[0]) for k in range(2, x.bit_length()))+sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
return bisection(f, n, n)**2 # Chai Wah Wu, Feb 08 2025
CROSSREFS
Cf. A059404, A126706, A177492 (k^2 for k in A120944), A286708, A359280, A362605, A378768 (k^2 for k in A286708).
Sequence in context: A154051 A335543 A366854 * A217584 A030633 A189988
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Feb 06 2025
STATUS
approved