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A367931 a(n) is the smallest number k such that k*n is an exponentially odious number (A270428).

%I A367931 #10 Dec 05 2023 08:54:34
%S A367931 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,3,1,1,1,1,4,1,1,
%T A367931 1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,2,1,1,1,1,1,1,1,2,1,1,1,1,
%U A367931 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N A367931 a(n) is the smallest number k such that k*n is an exponentially odious number (A270428).
%H A367931 Amiram Eldar, <a href="/A367931/b367931.txt">Table of n, a(n) for n = 1..10000</a>
%F A367931 Multiplicative with a(p^e) = p^s(e), where s(e) = min{k >= e, k is odious} - e.
%F A367931 a(n) = A367933(n)/n.
%F A367931 a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).
%F A367931 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.30023300..., where f(x) = (1-x) * (1 + Sum_{k>=1} x*(k-s(k))), and s(k) is defined above.
%t A367931 f[p_, e_] := Module[{k = e}, While[! OddQ[DigitCount[k, 2 ,1]], k++]; p^(k-e)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A367931 (PARI) s(e) = {my(k = e); while(!(hammingweight(k)%2), k++); k - e; };
%o A367931 a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]));}
%Y A367931 Cf. A000069, A270428, A367933.
%Y A367931 Similar sequences: A365298, A365685, A367932.
%K A367931 nonn,easy,mult,base
%O A367931 1,8
%A A367931 _Amiram Eldar_, Dec 05 2023