A357762 - OEIS

%I #7 Oct 12 2022 11:39:44

%S 1,1,9,6,2,8,3,2,6,4,3,2,5,2,5,6,4,3,7,2,2,2,2,9,1,6,3,3,2,0,0,8,1,9,

%T 1,8,1,0,1,0,4,2,6,7,4,6,4,0,1,5,9,4,3,8,1,8,9,8,7,2,3,3,3,7,3,0,7,8,

%U 3,7,5,1,6,1,0,9,1,5,8,0,8,7,7,7,9,1,1,9,6,4,5,4,6,2,1,1,0,7,4,8,9,6,3,3,3

%N Decimal expansion of -Sum_{k>=1} A106400(k)/k.

%C The asymptotic mean of the excess of the number of odious divisors over the number of evil divisors (A357761, see formula).

%C The convergence of the partial sums S(m) = -Sum_{k=1..2^m-1} A106400(k)/k is fast: e.g., S(28) is already correct to 100 decimal digits (see also _Jon E. Schoenfield_'s comment in A351404).

%F Equals -2 * Sum_{k>=1} A106400(2*k-1)/(2*k-1).

%F Equals lim_{m->oo} (1/m) * Sum_{k=1..m} A357761(k).

%e 1.19628326432525643722229163320081918101042674640159...

%t sum = 0; m = 1; pow = 2; Do[sum -= (-1)^DigitCount[k, 2, 1]/k; If[k == pow - 1, Print[m, " ", N[sum, 120]]; m++; pow *= 2], {k, 1, 2^30}]

%o (PARI) default(realprecision, 150);

%o sm = 0.; m = 1; pow = 2; for(k = 1, 2^30, sm -= (-1)^hammingweight(k)/k; if(k == pow - 1, print(m," ",sm); m++; pow *= 2))

%Y Cf. A106400, A357761.

%Y Similar constants: A215016, A351404

%K nonn,cons

%O 1,3

%A _Amiram Eldar_, Oct 12 2022