cp's OEIS Frontend

A384557 The number of exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

%I A384557 #10 Jun 03 2025 11:57:16
%S A384557 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,2,1,1,
%T A384557 1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,2,1,1,1,1,
%U A384557 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1
%N A384557 The number of exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).
%C A384557 First differs from A359411 at n = 2097152 = 2^21: a(2097152) = 4, while A359411(2097152) = 2.
%C A384557 First differs from A368979 at n = 512 = 2^9: a(512) = 2, while A368979(512) = 3.
%C A384557 First differs from A367516 at n = 128 = 2^7: a(128) = 2, while A367516(128) = 1.
%C A384557 First differs from A382291 at n = 128 = 2^7: a(128) = 2, while A382291(128) = 4.
%C A384557 First differs from A368168 at n = 64 = 2^6: a(64) = 2, while A368168(64) = 1.
%C A384557 The sum of these divisors is A384559(n), and the largest of them is A331737(n).
%C A384557 The number of exponential unitary (or e-unitary) divisors of n is A278908(n) and the number of divisors of n that are exponentially odd numbers is A322483(n).
%C A384557 All the terms are powers of 2. The first term that is greater than 2 is a(32768) = 4.
%H A384557 Amiram Eldar, <a href="/A384557/b384557.txt">Table of n, a(n) for n = 1..10000</a>
%F A384557 Multiplicative with a(p^e) = A068068(e).
%F A384557 a(n) >= 1, with equality if and only if n is in A138302.
%F A384557 a(n) <= A278908(n), with equality if and only if n is an exponentially odd number.
%F A384557 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=2} (d(k) - d(k-1))/p^k) = 1.13551542615965557947..., where d(k) is the number of odd unitary divisors of k (A068068).
%t A384557 f[p_, e_] := 2^PrimeNu[e/2^IntegerExponent[e, 2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384557 (PARI) a(n) = vecprod(apply(x -> 2^omega(x >> valuation(x, 2)) , factor(n)[, 2]));
%Y A384557 Cf. A068068, A138302, A268335, A278908, A322483, A331737, A359411, A367516, A368168, A368979, A382291, A384559.
%K A384557 nonn,easy,mult
%O A384557 1,8
%A A384557 _Amiram Eldar_, Jun 03 2025