A382065 Exponentially refactorable numbers: numbers whose exponents in their canonical prime factorization are all refactorable numbers (A033950).

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Author

Amiram Eldar, Mar 14 2025

Keywords

• nonn
• easy

Comments

First differs from A377019 at n = 55: A377019(55) = 64 is not a term of this sequence.

First differs from A344742 at n = 62: A344742(62) = 72 is not a term of this sequence.

All the cubefree numbers (A004709) are terms. The least term that is not cubefree is a(215) = 256 = 2^8.

Subsequence of A382063 and first differs from it at n = 362: A382063(362) = 432 = 2^4 * 3^3 is not a term of this sequence.

The asymptotic density of this sequence is Product_{p prime} (1 - 1/p^3 + (1 - 1/p) * (Sum_{k>=3} 1/p^A033950(k))) = 0.83493143539605138255... .

The relative density of this sequence within A382063 is the ratio between the densities of the two sequences: 0.997553... .

Links

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

Crossrefs

Cf. A033950, A344742.

Subsequence of A382063.

Subsequence: A004709.

Similar sequences: A197680, A209061, A138302, A268335, A361177, A377019.

Programs

Mathematica

refQ[k_] := Divisible[k, DivisorSigma[0, k]]; q[k_] := AllTrue[FactorInteger[k][[;; , 2]], refQ]; Select[Range[100], q]

PARI

isref(n) = !(n % numdiv(n));
isok(k) = {my(e = factor(k)[, 2]); for(i = 1, #e, if(!isref(e[i]), return(0))); 1; }