cp's OEIS Frontend

A119241 Number of powerful numbers (A001694) between consecutive squares n^2 and (n+1)^2.

%I A119241 #14 Sep 10 2024 11:03:42
%S A119241 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 A119241 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 A119241 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 A119241 Number of powerful numbers (A001694) between consecutive squares n^2 and (n+1)^2.
%C A119241 Is there an upper bound on the number of powerful numbers between consecutive squares? Pettigrew conjectures that there is no bound. See A119242.
%C A119241 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 A119241 József Sándor, Dragoslav S. Mitrinovic and Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, chapter VI, p. 226.
%H A119241 Amiram Eldar, <a href="/A119241/b119241.txt">Table of n, a(n) for n = 1..10000</a>
%H A119241 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 A119241 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 A119241 a(5) = 2 because the two powerful numbers 27 and 32 are between 25 and 36.
%t A119241 Powerful[n_] := (n==1) || Min[Transpose[FactorInteger[n]][[2]]]>1; Table[Length[Select[Range[k^2+1, k^2+2k], Powerful[ # ]&]], {k,130}]
%o A119241 (Python)
%o A119241 from math import isqrt
%o A119241 from sympy import integer_nthroot, factorint
%o A119241 def A119241(n):
%o A119241     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 A119241     return f((n+1)**2-1)-f(n**2) # _Chai Wah Wu_, Sep 10 2024
%Y A119241 Cf. A001694, A090699, A119242.
%K A119241 nonn
%O A119241 1,5
%A A119241 _T. D. Noe_, May 09 2006