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

1,1

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

Mathematica

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

Prog

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

Crossrefs

Equals A356862 \ A246655.

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

Sequence in context: A359890 A323055 A059404 * A303946 A360246 A242416

Adjacent sequences: A376247 A376248 A376249 * A376251 A376252 A376253

Keyword

nonn,easy

Author

Amiram Eldar, Sep 17 2024

Status

approved