A381824 Odd cubefull numbers: odd numbers that are divisible by the cube of any of their prime factors.

1, 27, 81, 125, 243, 343, 625, 729, 1331, 2187, 2197, 2401, 3125, 3375, 4913, 6561, 6859, 9261, 10125, 12167, 14641, 15625, 16807, 16875, 19683, 24389, 27783, 28561, 29791, 30375, 35937, 42875, 50625, 50653, 59049, 59319, 64827, 68921, 78125, 79507, 83349, 83521, 84375, 91125

Offset: 1

Author

Amiram Eldar, Mar 08 2025

Keywords

• nonn
• easy

Comments

Numbers whose prime factorization has primes and exponents that are larger than 2 (except for 1 whose prime factorization is empty).

Numbers k such that A020639(k) >= 3 and A051904(k) >= 3.

Links

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

Crossrefs

Intersection of A005408 and A036966.

Subsequences: A016755 (odd cubes), A381825 (odd cubefull exponentially odd numbers).

Cf. A020639, A051904, A065483.

Programs

Mathematica

Join[{1}, Select[Range[3, 10000, 2], Min[FactorInteger[#][[;; , 2]]] > 2 &]]

PARI

isok(k) = k == 1 || (k % 2 && vecmin(factor(k)[, 2]) > 2);

Formula

Sum_{n>=1} 1/a(n) = Product_{p prime >= 3} (1 + 1/(p^2*(p-1))) = (4/5) * A065483 = 1.07182732285947779727... .