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

1,3

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))

Crossrefs

Cf. A106400, A357761.

Similar constants: A215016, A351404

Sequence in context: A194182 A019961 A327996 * A093540 A252837 A198573

Adjacent sequences: A357759 A357760 A357761 * A357763 A357764 A357765

Keyword

nonn,cons

Author

Amiram Eldar, Oct 12 2022

Status

approved