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A367516 The number of unitary divisors of n that are exponentially evil numbers (A262675).

%I A367516 #11 Nov 23 2023 15:16:26
%S A367516 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 A367516 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 A367516 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A367516 The number of unitary divisors of n that are exponentially evil numbers (A262675).
%C A367516 First differs from A359411 at n = 128.
%H A367516 Amiram Eldar, <a href="/A367516/b367516.txt">Table of n, a(n) for n = 1..10000</a>
%F A367516 Multiplicative with a(p^e) = (2-A010060(e)).
%F A367516 a(n) = A034444(n)/A367515(n).
%F A367516 a(n) = 2^A367512(n).
%F A367516 a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).
%F A367516 a(n) <= A034444(n), with equality if and only if n is an exponentially evil number (A262675).
%F A367516 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.13071730542774788785..., where f(x) = 1/2 + x + ((1-x)/2) * Product_{k>=0} (1 - x^(2^k)).
%t A367516 f[p_, e_] := If[EvenQ[DigitCount[e, 2, 1]], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A367516 (PARI) a(n) = vecprod(apply(x -> 2-hammingweight(x)%2, factor(n)[, 2]));
%o A367516 (Python)
%o A367516 from sympy import factorint
%o A367516 def A367516(n): return 1<<sum(1 for e in factorint(n).values() if e.bit_count()&1^1) # _Chai Wah Wu_, Nov 23 2023
%Y A367516 Cf. A010060, A262675, A270428, A359411, A367512.
%Y A367516 Similar sequences: A034444, A055076, A056624, A366901, A366902, A367515.
%K A367516 nonn,easy,mult
%O A367516 1,8
%A A367516 _Amiram Eldar_, Nov 21 2023