A372695 Cubefull numbers that are not prime powers.

216, 432, 648, 864, 1000, 1296, 1728, 1944, 2000, 2592, 2744, 3375, 3456, 3888, 4000, 5000, 5184, 5488, 5832, 6912, 7776, 8000, 9261, 10000, 10125, 10368, 10648, 10976, 11664, 13824, 15552, 16000, 16875, 17496, 17576, 19208, 20000, 20736, 21296, 21952, 23328, 25000

Offset

1,1

Comments

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.

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

Superset of A372841.

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

Intersection of A036966 and A024619.

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

Example

Table of smallest 12 terms and instances of omega(a(n)) = m for m = 2..4
    n      a(n)
  ------------------------
    1      216 = 2^3 * 3^3
    2      432 = 2^4 * 3^3
    3      648 = 2^3 * 3^4
    4      864 = 2^5 * 3^3
    5     1000 = 2^3 * 5^3
    6     1296 = 2^4 * 3^4
    7     1728 = 2^6 * 3^3
    8     1944 = 2^3 * 3^5
    9     2000 = 2^4 * 5^3
   10     2592 = 2^5 * 3^4
   11     2744 = 2^3 * 7^3
   12     3375 = 3^3 * 5^3
  ...
   43    27000 = 2^3 * 3^3 * 5^3
  ...
  587  9261000 = 2^3 * 3^3 * 5^3 * 7^3

Mathematica

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]

Prog

(Python)
from math import gcd
from sympy import primepi, integer_nthroot, factorint
def A372695(n):
    def f(x):
        c = n+1+x+sum(primepi(integer_nthroot(x, k)[0]) for k in range(3, x.bit_length()))
        for w in range(1, integer_nthroot(x, 5)[0]+1):
            if all(d<=1 for d in factorint(w).values()):
                for y in range(1, integer_nthroot(z:=x//w**5, 4)[0]+1):
                    if gcd(w, y)==1 and all(d<=1 for d in factorint(y).values()):
                        c -= integer_nthroot(z//y**4, 3)[0]
        return c
    def bisection(f, kmin=0, kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f, n, n) # Chai Wah Wu, Sep 12 2024

Crossrefs

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

Cf. A065483, A152441.

Sequence in context: A339245 A327284 A376936 * A043371 A304369 A074266

Adjacent sequences: A372692 A372693 A372694 * A372696 A372697 A372698

Keyword

nonn

Author

Michael De Vlieger, May 14 2024

Status

approved