A376250 - OEIS

%I #6 Sep 20 2024 06:11:35

%S 12,18,20,24,28,40,44,45,48,50,52,54,56,60,63,68,72,75,76,80,84,88,90,

%T 92,96,98,99,104,108,112,116,117,120,124,126,132,135,136,140,144,147,

%U 148,150,152,153,156,160,162,164,168,171,172,175,176,184,188,189,192,198,200

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

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

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

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

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

%C All the factorial numbers above 6 are terms.

%C 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.

%H Amiram Eldar, <a href="/A376250/b376250.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%Y Equals A356862 \ A246655.

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

%K nonn,easy

%O 1,1

%A _Amiram Eldar_, Sep 17 2024