OFFSET
1,1
COMMENTS
Numbers such that all prime power factor exponents exceed 2, and at least one, but not all of the exponents exceed 3.
Numbers whose smallest prime exponent is 3 and whose largest prime exponent exceeds 3. - Antti Karttunen, Jan 22 2026
This sequence is A036966 \ A036967 \ A062838, that is, cubefull numbers without biquadratefull numbers and cubes of squarefree numbers.
Analogous to A325240 (squareful numbers that are not cubefull).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
EXAMPLE
Table of n, a(n) for select n:
n a(n)
----------------------------
1 432 = 2^4 * 3^3
2 648 = 2^3 * 3^4
3 864 = 2^5 * 3^3
4 1728 = 2^6 * 3^3
5 1944 = 2^3 * 3^5
6 2000 = 2^4 * 5^3
7 3456 = 2^7 * 3^3
8 4000 = 2^5 * 5^3
9 5000 = 2^3 * 5^4
10 5488 = 2^4 * 7^3
14 10125 = 3^4 * 5^3
32 54000 = 2^4 * 3^3 * 5^3
MATHEMATICA
nn = 10^5; Union@ Flatten@ Table[If[0 < Count[FactorInteger[#][[;; , -1]], _?(# > 3 &)] < PrimeNu[#], #, Nothing] &[a^5 * b^4 * c^3], {c, Surd[nn, 3]}, {b, Surd[nn/(c^3), 4]}, {a, Surd[nn/(b^4 * c^3), 5]}]
PROG
(PARI) is_A391593(n) = if(n<=1, 0, (e->(vecmin(e)==3 && vecmax(e)>3))(factor(n)[, 2])); \\ Antti Karttunen, Jan 22 2026
(Python)
from math import isqrt, gcd
from sympy import mobius, integer_nthroot, factorint
from oeis_sequences.OEISsequences import bisection
def A391593(n):
def f(x):
x3 = integer_nthroot(x, 3)[0]
c = n+x-1+sum(mobius(k)*(x3//k**2) for k in range(1, isqrt(x3)+1))
for t in range(1, integer_nthroot(x, 5)[0]+1):
if all(d<=1 for d in factorint(t).values()):
for y in range(1, integer_nthroot(v:=x//t**5, 4)[0]+1):
if gcd(t, y)==1 and all(d<=1 for d in factorint(y).values()):
c -= integer_nthroot(v//y**4, 3)[0]
for u in range(1, integer_nthroot(x, 7)[0]+1):
if all(d<=1 for d in factorint(u).values()):
for w in range(1, integer_nthroot(a:=x//u**7, 6)[0]+1):
if gcd(w, u)==1 and all(d<=1 for d in factorint(w).values()):
for y in range(1, integer_nthroot(z:=a//w**6, 5)[0]+1):
if gcd(w, y)==1 and gcd(u, y)==1 and all(d<=1 for d in factorint(y).values()):
c += integer_nthroot(z//y**5, 4)[0]
return c
return bisection(f, n, n) # Chai Wah Wu, Jan 29 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Jan 13 2026
STATUS
approved