A372695 - OEIS

%I #20 Sep 13 2024 08:04:04

%S 216,432,648,864,1000,1296,1728,1944,2000,2592,2744,3375,3456,3888,

%T 4000,5000,5184,5488,5832,6912,7776,8000,9261,10000,10125,10368,10648,

%U 10976,11664,13824,15552,16000,16875,17496,17576,19208,20000,20736,21296,21952,23328,25000

%N Cubefull numbers that are not prime powers.

%C Numbers k such that rad(k)^3 | k and omega(k) > 1. In other words, numbers with at least 2 distinct prime factors whose prime power factors have exponents that exceed 2.

%C Proper subset of the following sequences: A001694, A036966, A126706, A286708.

%C Superset of A372841.

%C Smallest term k with omega(k) = m is k = A002110(m)^3 = A115964(m).

%H Michael De Vlieger, <a href="/A372695/b372695.txt">Table of n, a(n) for n = 1..10000</a>

%F Intersection of A036966 and A024619.

%F Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p^2*(p-1))) - Sum_{p prime} 1/(p^2*(p-1)) - 1 = A065483 - A152441 - 1 = 0.0188749045... . - _Amiram Eldar_, May 17 2024

%e Table of smallest 12 terms and instances of omega(a(n)) = m for m = 2..4

%e     n      a(n)

%e   ------------------------

%e     1      216 = 2^3 * 3^3

%e     2      432 = 2^4 * 3^3

%e     3      648 = 2^3 * 3^4

%e     4      864 = 2^5 * 3^3

%e     5     1000 = 2^3 * 5^3

%e     6     1296 = 2^4 * 3^4

%e     7     1728 = 2^6 * 3^3

%e     8     1944 = 2^3 * 3^5

%e     9     2000 = 2^4 * 5^3

%e    10     2592 = 2^5 * 3^4

%e    11     2744 = 2^3 * 7^3

%e    12     3375 = 3^3 * 5^3

%e   ...

%e    43    27000 = 2^3 * 3^3 * 5^3

%e   ...

%e   587  9261000 = 2^3 * 3^3 * 5^3 * 7^3

%t nn = 25000; Rest@ Select[Union@ Flatten@ Table[a^5 * b^4 * c^3, {c, Surd[nn, 3]}, {b, Surd[nn/(c^3), 4]}, {a, Surd[nn/(b^4 * c^3), 5]}], Not@*PrimePowerQ]

%o (Python)

%o from math import gcd

%o from sympy import primepi, integer_nthroot, factorint

%o def A372695(n):

%o     def f(x):

%o         c = n+1+x+sum(primepi(integer_nthroot(x, k)[0]) for k in range(3, x.bit_length()))

%o         for w in range(1,integer_nthroot(x,5)[0]+1):

%o             if all(d<=1 for d in factorint(w).values()):

%o                 for y in range(1,integer_nthroot(z:=x//w**5,4)[0]+1):

%o                     if gcd(w,y)==1 and all(d<=1 for d in factorint(y).values()):

%o                         c -= integer_nthroot(z//y**4,3)[0]

%o         return c

%o     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     return bisection(f,n,n) # _Chai Wah Wu_, Sep 12 2024

%Y Cf. A001694, A007947, A024619, A036966, A115964, A126706, A286708, A372841.

%Y Cf. A065483, A152441.

%K nonn

%O 1,1

%A _Michael De Vlieger_, May 14 2024