A357762 - OEIS

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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

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OFFSET

1,3

COMMENTS

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).

LINKS

Table of n, a(n) for n=1..105.

FORMULA

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

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

EXAMPLE

1.19628326432525643722229163320081918101042674640159...

MATHEMATICA

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}]

PROG

(PARI) 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))