A367698 - OEIS

%I #11 Nov 27 2023 09:48:10

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

%T 1,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,3,1,2,1,1,1,1,1,1,1,4,1,1,1,1,

%U 1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1

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

%C First differs from A055229 at n = 64.

%H Amiram Eldar, <a href="/A367698/b367698.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = n/A366905(n).

%F Multiplicative with a(p^e) = p^(e-s(e)), where s(e) = max({k=1..e, k odious}).

%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) = 1.25857819194624249136..., 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 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 (PARI) s(n) = {my(k = n); while(!(hammingweight(k)%2), k--); n-k; }

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

%Y Cf. A055229, A270428, A366905, A367699.

%K nonn,easy,mult,base

%O 1,8

%A _Amiram Eldar_, Nov 27 2023