A390439 - OEIS

%I #18 Dec 10 2025 13:34:49

%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,19,20,21,22,23,24,25,26,27,

%T 28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,49,50,51,

%U 52,53,54,55,56,57,58,59,60,61,62,63,65,66,67,68,69,70,71,72

%N Exponentially noncomposite numbers: numbers whose prime factorization exponents are all noncomposites.

%C Subsequence of A209061, A386803 and A388972, and first differs from them at n = 62: A209061(62) = A386803(62) = A388972(62) = 64 = 2^6 is not a term in this sequence.

%C First differs from A115063 and A369939 at n = 122: a(122) = 128 = 2^7 is neither a term in A115063 nor a term in A369939. Also, A115063(243) = A369939(243) = 256 = 2^8 is the least term in A115063 and A369939 that is not a term in this sequence.

%C Except for a(1) = 1, a subsequence of A140823 and differs from it by not having the terms 48, 64, 80, 112, 144, ... .

%C The squarefree numbers (A005117) and numbers whose prime factorization exponents are all primes (A056166) are all terms in this sequence. Each term in this sequence has a unique representation as the product of two coprime numbers, one is squarefree and the other is a term in A056166.

%C The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^4 + Sum_{q prime >= 5} (1/p^q - 1/p^(q+1))) = 0.94671933735527801046... . [corrected by _Jason Yuen_, Nov 11 2025]

%C Numbers for which their sets of exponential divisors (A322791), exponential unitary divisors (A361255), and exponential semiproper divisors (defined in A323308 and A323309) coincide. - _Amiram Eldar_, Dec 05 2025

%H Amiram Eldar, <a href="/A390439/b390439.txt">Table of n, a(n) for n = 1..10000</a>

%t q[n_] := AllTrue[FactorInteger[n][[;; , 2]], ! CompositeQ[#] &]; Select[Range[100], q]

%o (PARI) isok(k) = vecsum(apply(x -> if(x == 1 || isprime(x), 0, 1), factor(k)[, 2])) == 0;

%Y Subsequence of A209061, A386803 and A388972.

%Y Subsequences: A005117, A004709, A046100, A056166.

%Y Cf. A008578, A115063, A140823, A274034, A369939.

%Y Cf. A322791, A323308, A323309, A361255.

%K nonn,easy

%O 1,2

%A _Amiram Eldar_, Nov 05 2025