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A335988
Cubefull exponentially odd numbers: numbers whose prime factorization contains only odd exponents that are larger than 1.
21
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
1,2
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.
LINKS
FORMULA
Sum_{n>=1} 1/a(n) = Product_{p prime} (1 + 1/(p*(p^2-1))) = 1.2312911... (A065487).
EXAMPLE
8 = 2^3 is a term since the exponent of its prime factor 2 is 3 which is odd and larger than 1.
MATHEMATICA
Join[{1}, Select[Range[10^5], AllTrue[Last /@ FactorInteger[#], #1 > 1 && OddQ[#1] &] &]]
PROG
(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
CROSSREFS
Intersection of A001694 and A268335.
Intersection of A036966 and A268335.
A355038 in ascending order.
A030078, A050997, A092759, A179665, A079395 and A138031 are subsequences.
Sequence in context: A116002 A339595 A376173 * A097054 A370788 A304291
KEYWORD
nonn
AUTHOR
Amiram Eldar, Jul 03 2020
STATUS
approved