A336177 Numbers k such that there are exactly two powerful numbers between k^2 and (k+1)^2.

5, 11, 14, 22, 25, 33, 44, 46, 55, 58, 62, 70, 72, 73, 82, 88, 96, 98, 103, 104, 109, 110, 111, 124, 129, 135, 155, 156, 158, 164, 172, 176, 178, 181, 187, 197, 203, 206, 207, 209, 212, 218, 240, 243, 248, 249, 254, 257, 259, 268, 277, 279, 281, 285, 288, 291

Offset

1,1

Comments

Positions of 2's in A119241.

Shiu (1980) proved that this sequence has an asymptotic density 0.2312... A more accurate calculation using his formula gives 0.231299167354828...

References

József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.

Links

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

P. Shiu, On the number of square-full integers between successive squares, Mathematika, Vol. 27, No. 2 (1980), pp. 171-178.

Example

5 is a term since there are exactly two powerful numbers, 27 = 3^3 and 32 = 2^5 between 5^2 = 25 and (5+1)^2 = 36.

Mathematica

powQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[300], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?powQ] == 2 &]

Prog

(Python)
from functools import lru_cache
from math import isqrt
from sympy import mobius, integer_nthroot
from oeis_sequences.OEISsequences import bisection
def A336177(n):
    @lru_cache(maxsize=None)
    def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
    @lru_cache(maxsize=None)
    def g(x):
        c, l = 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)
        c += squarefreepi(integer_nthroot(x, 3)[0])-l
        return c
    def f(x):
        c, a = n+x, 1
        for k in range(1, x+1):
            b = g((k+1)**2)
            if b == a+3:
                c -= 1
            a = b
        return c
    return bisection(f, n, n) # Chai Wah Wu, Mar 14 2026

Crossrefs

Cf. A001694, A119241, A119242, A336175, A336176, A336178.

Sequence in context: A267298 A102801 A314000 * A314001 A196206 A079030

Adjacent sequences: A336174 A336175 A336176 * A336178 A336179 A336180

Keyword

nonn

Author

Amiram Eldar, Jul 10 2020

Status

approved