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A366901 The number of exponentially odious divisors of n.

%I A366901 #11 Feb 24 2024 16:16:24
%S A366901 1,2,2,3,2,4,2,3,3,4,2,6,2,4,4,4,2,6,2,6,4,4,2,6,3,4,3,6,2,8,2,4,4,4,
%T A366901 4,9,2,4,4,6,2,8,2,6,6,4,2,8,3,6,4,6,2,6,4,6,4,4,2,12,2,4,6,4,4,8,2,6,
%U A366901 4,8,2,9,2,4,6,6,4,8,2,8,4,4,2,12,4,4,4
%N A366901 The number of exponentially odious divisors of n.
%C A366901 First differs from A049599 and A282446 at n = 32, from A365551 at n = 64, and from A353898 at n = 128.
%C A366901 The number of divisors of n that are exponentially odious numbers (A270428), i.e., numbers having only odious (A000069) exponents in their canonical prime factorization.
%C A366901 The sum of these divisors is A366903(n) and the largest of them is A366905(n).
%H A366901 Amiram Eldar, <a href="/A366901/b366901.txt">Table of n, a(n) for n = 1..10000</a>
%F A366901 Multiplicative with a(p^e) = A115384(e) + 1.
%F A366901 a(n) <= A000005(n), with equality if and only if n is a cubefree number (A004709).
%t A366901 f[p_, e_] := Floor[e/2] + If[OddQ[e] || EvenQ[DigitCount[e + 1, 2, 1]], 1, 0] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A366901 (PARI) s(n) = 1 + n\2 + (n%2 || hammingweight(n+1)%2==0); \\ after _Charles R Greathouse IV_ at A115384
%o A366901 a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));
%Y A366901 Cf. A000005, A000069, A004709, A115384, A270428, A366903, A366905.
%Y A366901 Cf. A049599, A282446, A365551, A353898.
%Y A366901 Similar sequences: A325837, A353898, A365680, A366902.
%K A366901 nonn,easy,mult
%O A366901 1,2
%A A366901 _Amiram Eldar_, Oct 27 2023