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A369118 k is 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. 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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)
LINKS
EXAMPLE
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.
MATHEMATICA
A369118Q[n_] := OddQ[n] && AllTrue[FactorInteger[n], OddQ[#] && PrimeQ[#]&, 2];
Select[Range[500000], A369118Q] (* Paolo Xausa, Jan 19 2024 *)
PROG
(SageMath)
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)])
CROSSREFS
Cf. A002808 (superset), A001694 (superset).
A051674 is a subsequence for n>1.
Sequence in context: A137800 A125497 A179145 * A118092 A371189 A126272
KEYWORD
nonn
AUTHOR
Peter Luschny, Jan 17 2024
STATUS
approved

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Last modified July 12 19:59 EDT 2024. Contains 374252 sequences. (Running on oeis4.)