A367516 - OEIS

%I #11 Nov 23 2023 15:16:26

%S 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 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 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

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

%C First differs from A359411 at n = 128.

%H Amiram Eldar, <a href="/A367516/b367516.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = (2-A010060(e)).

%F a(n) = A034444(n)/A367515(n).

%F a(n) = 2^A367512(n).

%F a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).

%F a(n) <= A034444(n), with equality if and only if n is an exponentially evil number (A262675).

%F 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 f[p_, e_] := If[EvenQ[DigitCount[e, 2, 1]], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecprod(apply(x -> 2-hammingweight(x)%2, factor(n)[, 2]));

%o (Python)

%o from sympy import factorint

%o 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 Cf. A010060, A262675, A270428, A359411, A367512.

%Y Similar sequences: A034444, A055076, A056624, A366901, A366902, A367515.

%K nonn,easy,mult

%O 1,8

%A _Amiram Eldar_, Nov 21 2023