A396594 Numbers m such that rad(m) * bigomega(m) = m.

2, 3, 4, 5, 7, 11, 13, 17, 18, 19, 23, 24, 29, 31, 37, 40, 41, 43, 45, 47, 53, 56, 59, 61, 63, 67, 71, 73, 79, 83, 88, 89, 97, 99, 101, 103, 104, 107, 109, 113, 117, 127, 131, 136, 137, 139, 149, 151, 152, 153, 157, 163, 167, 171, 173, 179, 181, 184, 191, 193

Offset

1,1

Comments

Primes p are terms since rad(p) * bigomega(p) = p * 1 = p.

From Michael De Vlieger, Jul 03 2026: (Start)

Primes are the only squarefree numbers (in A005117) in the sequence. Squarefree k = rad(k) such that bigomega(k) = omega(k) > 2 do not appear in the sequence since k * omega(k) > k.

The only powerful number k (in A001694) in this sequence is 4 = 2^2 = 2*2, since powerful numbers k imply rad(k)^2 | k, and further, bigomega(k) >= k/rad(k), where k/rad(k) >= rad(k). Attempting to increment the left hand side only increases the ratio RHS/LHS, since RHS increases by a prime factor.

Consequences:

1. A175787 is a proper subset of this sequence.

2. There is no intersection of this sequence and A120944 (squarefree and composite).

3. {a(n)} \ A175787 is a proper subset of A332785.

4. 4 is the only perfect power (in A001597) and the only prime power (in A246547) in this sequence. (End)

Links

James C. McMahon, Table of n, a(n) for n = 1..10000

Example

18 is a term since rad(18) * bigomega(18) = 6 * 3 = 18.

Mathematica

q[m_]:=PrimeOmega[m]*Times@@First/@FactorInteger[m]==m; Select[Range[193], q]

Crossrefs

Cf. A000040, A001222, A001597, A001694, A007947, A175787, A332785, A397221.

Sequence in context: A285304 A036844 A284696 * A033070 A211781 A348283

Adjacent sequences: A396590 A396592 A396593 * A396597 A396598 A396599

Keyword

nonn,new

Author

James C. McMahon and Vincenzo Manto, Jun 28 2026

Status

approved