A376250 Numbers with a unique largest prime exponent (A356862) that are not prime powers (A246655).

12, 18, 20, 24, 28, 40, 44, 45, 48, 50, 52, 54, 56, 60, 63, 68, 72, 75, 76, 80, 84, 88, 90, 92, 96, 98, 99, 104, 108, 112, 116, 117, 120, 124, 126, 132, 135, 136, 140, 144, 147, 148, 150, 152, 153, 156, 160, 162, 164, 168, 171, 172, 175, 176, 184, 188, 189, 192, 198, 200

Offset: 1

Author

Amiram Eldar, Sep 17 2024

Keywords

• nonn
• easy

Comments

First differs from A059404 at n = 55: A059404(55) = 180 = 2^2 * 3^2 * 5 is not a term of this sequence.

First differs from A360248 at n = 23: a(23) = 90 = 2 * 3^2 * 5 is not a term of A360248.

First differs from A332785 at n = 17: a(17) = 72 = 2^3 * 3^2 is not a term of A332785.

Numbers whose unordered prime signature (i.e., sorted, see A118914) ends with two different integers: {..., k, m} for some 1 <= k < m.

All the factorial numbers above 6 are terms.

The asymptotic density of this sequence is Sum_{k >= 1, p prime} (d(k+1, p) - d(k, p))/((p-1)*p^k) = 0.3660366524547281232052..., where d(k, p) = 0 for k = 1, and (1-1/p)/((1-1/p^k)*zeta(k)) for k > 1, is the density of terms that have in their prime factorization a prime p with the largest exponent that is > k.

Links

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

Crossrefs

Equals A356862 \ A246655.

Cf. A000142, A059404, A118914, A332785, A360248.

Programs

Mathematica

Select[Range[2, 200], Length[e = FactorInteger[#][[;; , 2]]] > 1 &&  Count[e, Max[e]] == 1 &]

PARI

is(k) = if (k == 1, 0, my(e = vecsort(factor(k)[,2])); #e > 1 && e[#e] > e[#e-1]);