

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
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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.
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



KEYWORD

nonn


AUTHOR



STATUS

approved



