cp's OEIS Frontend

A337736 The number of cubefull numbers (A036966) between the consecutive cubes n^3 and (n+1)^3.

%I A337736 #18 Sep 10 2024 17:26:04
%S A337736 0,1,1,1,1,2,1,2,1,2,0,4,2,1,3,0,3,1,2,1,3,2,0,2,5,1,3,1,1,3,3,2,1,3,
%T A337736 1,2,2,2,1,2,2,3,6,1,1,1,4,1,1,3,3,1,3,4,1,2,3,1,2,3,2,3,2,3,3,2,1,4,
%U A337736 2,1,1,0,7,1,1,4,3,2,2,2,3,3,2,0,4,2,4
%N A337736 The number of cubefull numbers (A036966) between the consecutive cubes n^3 and (n+1)^3.
%C A337736 For each k >= 0 the sequence of solutions to a(x) = k has a positive asymptotic density (Shiu, 1991).
%H A337736 Amiram Eldar, <a href="/A337736/b337736.txt">Table of n, a(n) for n = 1..10000</a>
%H A337736 P. Shiu, <a href="https://doi.org/10.1017/S0017089500008351">The distribution of cube-full numbers</a>, Glasgow Mathematical Journal, Vol. 33, No. 3 (1991), pp. 287-295. See section 3, p. 291.
%F A337736 Asymptotic mean: lim_{m->oo} (1/m) Sum_{k=1..m} a(k) = A362974 - 1 = 3.659266... . - _Amiram Eldar_, May 11 2023
%e A337736 a(2) = 1 since there is one cubefull number, 16 = 2^4, between 2^3 = 8 and 3^3 = 27.
%t A337736 cubQ[n_] := Min[FactorInteger[n][[;; , 2]]] > 2; a[n_] := Count[Range[n^3 + 1, (n + 1)^3 - 1], _?cubQ]; Array[a, 100]
%o A337736 (Python)
%o A337736 from math import gcd
%o A337736 from sympy import integer_nthroot, factorint
%o A337736 def A337736(n):
%o A337736     def f(x):
%o A337736         c = 0
%o A337736         for w in range(1,integer_nthroot(x,5)[0]+1):
%o A337736             if all(d<=1 for d in factorint(w).values()):
%o A337736                 for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):
%o A337736                     if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):
%o A337736                         c += integer_nthroot(z//y**4,3)[0]
%o A337736         return c
%o A337736     return f((n+1)**3-1)-f(n**3) # _Chai Wah Wu_, Sep 10 2024
%Y A337736 Cf. A000578, A036966, A119241, A337737, A362974.
%K A337736 nonn
%O A337736 1,6
%A A337736 _Amiram Eldar_, Sep 17 2020