A335988 Cubefull exponentially odd numbers: numbers whose prime factorization contains only odd exponents that are larger than 1.

1, 8, 27, 32, 125, 128, 216, 243, 343, 512, 864, 1000, 1331, 1944, 2048, 2187, 2197, 2744, 3125, 3375, 3456, 4000, 4913, 6859, 7776, 8192, 9261, 10648, 10976, 12167, 13824, 16000, 16807, 17496, 17576, 19683, 24389, 25000, 27000, 29791, 30375, 31104, 32768, 35937

Offset: 1

Author

Amiram Eldar, Jul 03 2020

Keywords

• nonn

Comments

This sequence is a permutation of A355038.

This sequence is also a permutation of the exponentially odd numbers (A268335) multiplied by the square of their squarefree kernel (A007947).

a(n)/rad(a(n)) is a permutation of the squares.

a(n)/rad(a(n))^2 is a permutation of the exponentially odd numbers.

Examples

			8 = 2^3 is a term since the exponent of its prime factor 2 is 3 which is odd and larger than 1.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Intersection of A001694 and A268335.

Intersection of A036966 and A268335.

A355038 in ascending order.

A030078, A050997, A092759, A179665, A079395 and A138031 are subsequences.

Cf. A007947, A065487.

Programs

Mathematica

Join[{1}, Select[Range[10^5], AllTrue[Last /@ FactorInteger[#], #1 > 1 && OddQ[#1] &] &]]

Python

from math import isqrt, prod
from sympy import factorint
def afind(N): # all terms up to limit N
    cands = (n**2*prod(factorint(n**2)) for n in range(1, isqrt(N//2)+2))
    return sorted(c for c in cands if c <= N)
print(afind(4*10**4)) # Michael S. Branicky, Jun 16 2022

Formula

Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.2312911... (A065487).