A367933 - OEIS

%I #8 Dec 05 2023 08:55:50

%S 1,2,3,4,5,6,7,16,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,48,25,

%T 26,81,28,29,30,31,128,33,34,35,36,37,38,39,80,41,42,43,44,45,46,47,

%U 48,49,50,51,52,53,162,55,112,57,58,59,60,61,62,63,128,65,66,67

%N a(n) is the smallest multiple of n that is an exponentially odious number (A270428).

%H Amiram Eldar, <a href="/A367933/b367933.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) = n * A367931(n).

%F a(n) >= n, with equality if and only if n is an exponentially odious number (A209061).

%F Sum_{k=1..n} a(k) ~  c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.30023300... , where f(x) = (1-x) * (1 + Sum_{k>=1} x^(2*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]; 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; };

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

%Y Cf. A270428, A367931.

%Y Similar sequences: A356192, A365684, A367934.

%K nonn,easy,mult,base

%O 1,2

%A _Amiram Eldar_, Dec 05 2023