A390439 Exponentially noncomposite numbers: numbers whose prime factorization exponents are all noncomposites.

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, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72

Offset

1,2

Comments

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.

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.

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

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.

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]

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

Links

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

Mathematica

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

Prog

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

Crossrefs

Subsequence of A209061, A386803 and A388972.

Subsequences: A005117, A004709, A046100, A056166.

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

Cf. A322791, A323308, A323309, A361255.

Sequence in context: A390438 A115063 A369939 * A178210 A013938 A339889

Adjacent sequences: A390436 A390437 A390438 * A390440 A390441 A390442

Keyword

nonn,easy

Author

Amiram Eldar, Nov 05 2025

Status

approved