cp's OEIS Frontend

A386258 Exponent of the highest power of 2 dividing the product of exponents of the prime factorization of n.

%I A386258 #11 Jul 18 2025 08:30:55
%S A386258 0,0,0,1,0,0,0,0,1,0,0,1,0,0,0,2,0,1,0,1,0,0,0,0,1,0,0,1,0,0,0,0,0,0,
%T A386258 0,2,0,0,0,0,0,0,0,1,1,0,0,2,1,1,0,1,0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,1,
%U A386258 0,0,0,1,0,0,1,1,0,0,0,2,2,0,0,1,0,0,0
%N A386258 Exponent of the highest power of 2 dividing the product of exponents of the prime factorization of n.
%C A386258 First differs from A386259 at n = 36.
%C A386258 First differs from A370078 at n = 64.
%C A386258 The first occurrence of k = 0, 1, 2, ... is at n = A085629(2^k) = 1, 4, 16, 144, 1296, 20736, 518400, ... .
%C A386258 The asymptotic density of the occurrences of 1 in this sequence is the asymptotic density of numbers whose prime factorization has only odd exponents except for one exponent that is of the form 4*k+2 (k >= 0) which equals A065463 * Sum_{p prime} p^2/(p^4+p^3+p-1) = 0.22670657681840536721... .
%H A386258 Amiram Eldar, <a href="/A386258/b386258.txt">Table of n, a(n) for n = 1..10000</a>
%F A386258 a(n) = A007814(A005361(n)).
%F A386258 Additive with a(p^e) = A007814(e).
%F A386258 a(n) = 0 if and only if n is an exponentially odd number (A268335).
%F A386258 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} f(1/p) = 0.37572872586497617473..., where f(x) = Sum_{k>=1} x^(2^k)/(1-x^(2^k)).
%t A386258 a[n_] := IntegerExponent[Times @@ FactorInteger[n][[;; , 2]], 2]; Array[a, 100]
%o A386258 (PARI) a(n) = vecsum(apply(x -> valuation(x, 2), factor(n)[, 2]));
%Y A386258 Cf. A005361, A007814, A065463, A085629, A268335, A295664, A370078, A386259.
%K A386258 nonn,easy
%O A386258 1,16
%A A386258 _Amiram Eldar_, Jul 17 2025