A367931 - OEIS

%I #10 Dec 05 2023 08:54:34

%S 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,3,1,1,1,1,4,1,1,

%T 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 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

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

%H Amiram Eldar, <a href="/A367931/b367931.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = p^s(e), where s(e) = min{k >= e, k is odious} - e.

%F a(n) = A367933(n)/n.

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

%F 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 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 (PARI) s(e) = {my(k = e); while(!(hammingweight(k)%2), k++); k - e; };

%o a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^s(f[i, 2]));}

%Y Cf. A000069, A270428, A367933.

%Y Similar sequences: A365298, A365685, A367932.

%K nonn,easy,mult,base

%O 1,8

%A _Amiram Eldar_, Dec 05 2023