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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.

%I A369118 #40 Mar 08 2025 08:47:04
%S A369118 27,125,243,343,1331,2187,2197,3125,3375,4913,6859,9261,12167,16807,
%T A369118 24389,29791,30375,35937,42875,50653,59319,68921,78125,79507,83349,
%U A369118 84375,103823,132651,148877,161051,166375,177147,185193,205379,226981,273375,274625
%N 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.
%C A369118 Every term is divisible by a cube.
%C A369118 If n and k are terms then n*k is a term if and only if gcd(n, k) = 1.
%C A369118 From _Michael De Vlieger_, Jan 19 2024: (Start)
%C A369118 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.
%C A369118 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.
%C A369118 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)
%H A369118 Michael De Vlieger, <a href="/A369118/b369118.txt">Table of n, a(n) for n = 1..10000</a>
%F A369118 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
%e A369118 25015118625 = 3^5 * 5^3 * 7^7 is a term.
%e A369118 3125 = 5^5 and 3375 = 3^3 * 5^3 are terms but 3125*3375 is not a term.
%t A369118 A369118Q[n_] := OddQ[n] && AllTrue[FactorInteger[n], OddQ[#] && PrimeQ[#]&, 2];
%t A369118 Select[Range[500000], A369118Q] (* _Paolo Xausa_, Jan 19 2024 *)
%o A369118 (SageMath)
%o A369118 def isA369118(n):
%o A369118     return (n > 1 and is_odd(n) and all(is_odd(f[1]) and is_prime(f[1])
%o A369118            for f in factor(n)))
%o A369118 print([n for n in range(1, 300000) if isA369118(n)])
%o A369118 (PARI) isok(k) = k > 1 && (k % 2 && #select(x -> (x <= 2) || !isprime(x), factor(k)[, 2]) == 0); \\ _Amiram Eldar_, Mar 08 2025
%Y A369118 Cf. A002808 (superset), A001694 (superset).
%Y A369118 A051674 is a subsequence for n>1.
%Y A369118 Intersects A001597, A013929, A052486, A246547, A286708.
%Y A369118 Subsequence of A381825.
%K A369118 nonn
%O A369118 1,1
%A A369118 _Peter Luschny_, Jan 17 2024