A381825 -id:A381825 - cp's OEIS frontend

A369118 k is a term if and only if k is a composite number where the bases and the exponents of its factors in the prime decomposition are all odd primes.

27, 125, 243, 343, 1331, 2187, 2197, 3125, 3375, 4913, 6859, 9261, 12167, 16807, 24389, 29791, 30375, 35937, 42875, 50653, 59319, 68921, 78125, 79507, 83349, 84375, 103823, 132651, 148877, 161051, 166375, 177147, 185193, 205379, 226981, 273375, 274625

Offset: 1

Peter Luschny, Jan 17 2024

nonn

Every term is divisible by a cube.
If n and k are terms then n*k is a term if and only if gcd(n, k) = 1.
From Michael De Vlieger, Jan 19 2024: (Start)
Prime exponents of prime power factors p^m | k imply that k is a powerful number. Hence this sequence is a proper subset of A001694, and k is of the form a^2 * b^3.
Prime exponents m imply either perfect powers k in A001597 such that all the m are the same, or an Achilles number k (in A052486) if the exponents differ. This is because prime p divides itself but is coprime to primes q != p. Therefore this sequence is not a subsequence of A001597.
The sequence consists of composite prime powers (A246547) and powerful numbers that are not prime powers (A286708), both of which are numbers that are not squarefree (A013929). (End)

			25015118625 = 3^5 * 5^3 * 7^7 is a term.
3125 = 5^5 and 3375 = 3^3 * 5^3 are terms but 3125*3375 is not a term.

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

Cf. A002808 (superset), A001694 (superset).
A051674 is a subsequence for n>1.
Intersects A001597, A013929, A052486, A246547, A286708.
Subsequence of A381825.

A369118Q[n_] := OddQ[n] && AllTrue[FactorInteger[n], OddQ[#] && PrimeQ[#]&, 2];
Select[Range[500000], A369118Q] (* Paolo Xausa, Jan 19 2024 *)

isok(k) = k > 1 && (k % 2 && #select(x -> (x <= 2) || !isprime(x), factor(k)[, 2]) == 0); \\ Amiram Eldar, Mar 08 2025

def isA369118(n):
    return (n > 1 and is_odd(n) and all(is_odd(f[1]) and is_prime(f[1])
           for f in factor(n)))
print([n for n in range(1, 300000) if isA369118(n)])

Sum_{n>=1} 1/a(n) = -1 + Product_{prime >= 3} (1 + Sum_{prime q >= 3} 1/p^q) = 0.05534030537711484966... . - Amiram Eldar, Mar 08 2025

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

Original entry on oeis.org

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

Amiram Eldar, Mar 08 2025

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.

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

Intersection of A005408 and A036966.
Subsequences: A016755 (odd cubes), A381825 (odd cubefull exponentially odd numbers).
Cf. A020639, A051904, A065483.

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

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

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

cp's OEIS Frontend

A369118 k is a term if and only if k is a composite number where the bases and the exponents of its factors in the prime decomposition are all odd primes.

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