cp's OEIS Frontend

A368979 The number of exponential divisors of n that are exponentially odd numbers (A268335).

%I A368979 #11 Jan 17 2024 04:25:29
%S A368979 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 A368979 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 A368979 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A368979 The number of exponential divisors of n that are exponentially odd numbers (A268335).
%C A368979 First differs from A367516 at n = 128, and from A359411 at n = 512.
%H A368979 Amiram Eldar, <a href="/A368979/b368979.txt">Table of n, a(n) for n = 1..10000</a>
%F A368979 Multiplicative with a(p^e) = A001227(e).
%F A368979 a(n) >= 1, with equality if and only if n is in A138302.
%F A368979 a(n) <= A049419(n), with equality if and only if n is noncomposite (A008578).
%F A368979 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) = 1.13657098749361390865..., where d(k) is the number of odd divisors of k (A001227).
%t A368979 f[p_, e_] := DivisorSigma[0, e/2^IntegerExponent[e, 2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368979 (PARI) a(n) = vecprod(apply(x -> numdiv(x >> valuation(x, 2)), factor(n)[, 2]));
%Y A368979 Cf. A001227, A008578, A049419, A138302, A268335, A322791, A368980.
%Y A368979 Cf. A359411, A367516.
%Y A368979 Similar sequences: A055076, A322483, A363825, A368977.
%K A368979 nonn,easy,mult
%O A368979 1,8
%A A368979 _Amiram Eldar_, Jan 11 2024