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A336178 Numbers k such that there are exactly three powerful numbers between k^2 and (k+1)^2.

%I A336178 #12 Sep 15 2024 22:01:48
%S A336178 31,36,67,93,132,140,145,161,166,189,192,220,223,265,280,290,296,311,
%T A336178 316,322,364,384,407,468,537,576,592,602,623,639,644,656,659,661,670,
%U A336178 690,722,769,771,793,828,883,888,890,896,950,961,981,984,987,992,995,1018
%N A336178 Numbers k such that there are exactly three powerful numbers between k^2 and (k+1)^2.
%C A336178 Positions of 3's in A119241.
%C A336178 Shiu (1980) proved that this sequence has an asymptotic density = 0.0770... A more accurate calculation using his formula gives 0.0770742722233...
%D A336178 József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.
%H A336178 Amiram Eldar, <a href="/A336178/b336178.txt">Table of n, a(n) for n = 1..10000</a>
%H A336178 P. Shiu, <a href="https://doi.org/10.1112/S0025579300010056">On the number of square-full integers between successive squares</a>, Mathematika, Vol. 27, No. 2 (1980), pp. 171-178.
%e A336178 31 is a term since there are exactly three powerful numbers, 968 = 2^3 * 11^2, 972 = 2^2 * 3^5 and 1000 = 2^3 * 5^3 between 31^2 = 961 and (31+1)^2 = 1024.
%t A336178 powQ[n_] := (n == 1) || Min @@ FactorInteger[n][[;; , 2]] > 1; Select[Range[1000], Count[Range[#^2 + 1, (# + 1)^2 - 1], _?powQ] == 3 &]
%o A336178 (Python)
%o A336178 from functools import lru_cache
%o A336178 from math import isqrt
%o A336178 from sympy import mobius, integer_nthroot
%o A336178 def A336178(n):
%o A336178     def squarefreepi(n): return int(sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)))
%o A336178     def bisection(f,kmin=0,kmax=1):
%o A336178         while f(kmax) > kmax: kmax <<= 1
%o A336178         while kmax-kmin > 1:
%o A336178             kmid = kmax+kmin>>1
%o A336178             if f(kmid) <= kmid:
%o A336178                 kmax = kmid
%o A336178             else:
%o A336178                 kmin = kmid
%o A336178         return kmax
%o A336178     @lru_cache(maxsize=None)
%o A336178     def g(x):
%o A336178         c, l = 0, 0
%o A336178         j = isqrt(x)
%o A336178         while j>1:
%o A336178             k2 = integer_nthroot(x//j**2,3)[0]+1
%o A336178             w = squarefreepi(k2-1)
%o A336178             c += j*(w-l)
%o A336178             l, j = w, isqrt(x//k2**3)
%o A336178         c += squarefreepi(integer_nthroot(x,3)[0])-l
%o A336178         return c
%o A336178     def f(x):
%o A336178         c, a = n+x, 1
%o A336178         for k in range(1,x+1):
%o A336178             b = g((k+1)**2)
%o A336178             if b == a+4:
%o A336178                 c -= 1
%o A336178             a = b
%o A336178         return c
%o A336178     return bisection(f,n,n) # _Chai Wah Wu_, Sep 14 2024
%Y A336178 Cf. A001694, A119241, A119242, A336175, A336176, A336177.
%K A336178 nonn
%O A336178 1,1
%A A336178 _Amiram Eldar_, Jul 10 2020