A351404 -id:A351404 - cp's OEIS frontend

A357762 Decimal expansion of -Sum_{k>=1} A106400(k)/k.

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, 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, 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

Offset: 1

Amiram Eldar, Oct 12 2022

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

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).

			1.19628326432525643722229163320081918101042674640159...

Cf. A106400, A357761.

Similar constants: A215016, A351404

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}]

default(realprecision, 150);
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))

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

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

cp's OEIS Frontend

A357762 Decimal expansion of -Sum_{k>=1} A106400(k)/k.

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