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Number of powerful numbers (A001694) between consecutive squares n^2 and (n+1)^2.
9

%I #14 Sep 10 2024 11:03:42

%S 0,1,0,0,2,0,0,1,0,1,2,0,0,2,1,1,0,1,1,1,0,2,0,0,2,0,0,1,1,0,3,0,2,0,

%T 0,3,1,0,1,0,1,1,0,2,1,2,0,1,0,1,1,1,1,0,2,1,1,2,1,0,0,2,1,0,1,0,3,0,

%U 0,2,0,2,2,1,0,1,1,1,1,0,0,2,1,1,0,0,1,2,1,1,0,1,3,1,0,2,0,2,0,1,1,1,2,2,0

%N Number of powerful numbers (A001694) between consecutive squares n^2 and (n+1)^2.

%C Is there an upper bound on the number of powerful numbers between consecutive squares? Pettigrew conjectures that there is no bound. See A119242.

%C This sequence is unbounded. For each k >= 0 the sequence of solutions to a(x) = k has a positive asymptotic density (Shiu, 1980). - _Amiram Eldar_, Jul 10 2020

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

%H Amiram Eldar, <a href="/A119241/b119241.txt">Table of n, a(n) for n = 1..10000</a>

%H 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.

%F Asymptotic mean: lim_{n->oo} (1/n) Sum_{k=1..n} a(k) = zeta(3/2)/zeta(3) - 1 = A090699 - 1 = 1.173254... - _Amiram Eldar_, Oct 24 2020

%e a(5) = 2 because the two powerful numbers 27 and 32 are between 25 and 36.

%t Powerful[n_] := (n==1) || Min[Transpose[FactorInteger[n]][[2]]]>1; Table[Length[Select[Range[k^2+1, k^2+2k], Powerful[ # ]&]], {k,130}]

%o (Python)

%o from math import isqrt

%o from sympy import integer_nthroot, factorint

%o def A119241(n):

%o def f(x): return int(sum(isqrt(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1) if all(d<=1 for d in factorint(k).values())))

%o return f((n+1)**2-1)-f(n**2) # _Chai Wah Wu_, Sep 10 2024

%Y Cf. A001694, A090699, A119242.

%K nonn

%O 1,5

%A _T. D. Noe_, May 09 2006