A371601 Nonsquarefree numbers whose largest nonunitary prime divisor is smaller than their smallest unitary prime divisor, if it exists.

4, 8, 9, 12, 16, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 52, 56, 60, 63, 64, 68, 72, 76, 80, 81, 84, 88, 92, 96, 99, 100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 128, 132, 135, 136, 140, 144, 148, 152, 153, 156, 160, 164, 168, 169, 171, 172, 175, 176

Offset: 1

Author

Amiram Eldar, Mar 29 2024

Keywords

• nonn
• easy

Comments

Subsequence of A283050 and first differs from it at n = 100: A283050(100) = 300 = 2^2 * 3 * 5^2 is not a term of this sequence.

Powerful numbers and nonpowerful numbers k such that 1 < A249740(k) < A277698(k), or equivalently, 1 < A006530(A057521(k)) < A020639(A055231(k)).

The asymptotic density of this sequence is (6/Pi^2) * Sum_{p prime} f(p)/(p^2-p+1) = 0.32131800923..., where f(p) = Product_{primes q <= p} (q^2-q+1)/(q^2-1).

Links

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

Crossrefs

Subsequence of A013929 and A283050.

Cf. A001694, A006530, A020639, A052485, A055231, A057521, A059956, A249740, A277698.

Programs

Mathematica

q[n_] := Module[{e = FactorInteger[n][[;; , 2]]}, Max[e] > 1 && (Min[e] > 1 || Max[e[[FirstPosition[e, 1][[1]] ;; -1]]] == 1)]; Select[Range[200], q]

PARI

is(n) = {my(e = apply(x->if(x > 1, 2, 1), factor(n)[,2])); n > 1 && vecmax(e) > 1 && vecsort(e, , 4) == e;}