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A367698 The smallest divisor d of n such that n/d is an exponentially odious number (A270428).

%I A367698 #11 Nov 27 2023 09:48:10
%S A367698 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,3,1,1,1,1,2,1,1,
%T A367698 1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,2,1,1,1,1,1,1,1,4,1,1,1,1,
%U A367698 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A367698 The smallest divisor d of n such that n/d is an exponentially odious number (A270428).
%C A367698 First differs from A055229 at n = 64.
%H A367698 Amiram Eldar, <a href="/A367698/b367698.txt">Table of n, a(n) for n = 1..10000</a>
%F A367698 a(n) = n/A366905(n).
%F A367698 Multiplicative with a(p^e) = p^(e-s(e)), where s(e) = max({k=1..e, k odious}).
%F A367698 a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).
%F A367698 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.25857819194624249136..., where f(x) = (1-x)*(1+Sum_{k>=1} x^s(k)), s(k) is defined above for k >= 1, and s(0) = 0.
%t A367698 maxOdious[e_] := Module[{k = e}, While[EvenQ[DigitCount[k, 2, 1]], k--]; k]; f[p_, e_] := p^(e - maxOdious[e]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A367698 (PARI) s(n) = {my(k = n); while(!(hammingweight(k)%2), k--); n-k; }
%o A367698 a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2])); }
%Y A367698 Cf. A055229, A270428, A366905, A367699.
%K A367698 nonn,easy,mult,base
%O A367698 1,8
%A A367698 _Amiram Eldar_, Nov 27 2023