A300723 Möbius-transform of A005187(A053645(n)).

0, 0, 1, 0, 1, 2, 4, 0, 0, 2, 4, 4, 8, 6, 9, 0, 1, 0, 4, 4, 3, 6, 11, 8, 15, 10, 18, 12, 23, 10, 26, 0, -4, 2, -1, 0, 8, 6, 2, 8, 16, 2, 19, 12, 12, 14, 26, 16, 28, 16, 33, 20, 39, 20, 37, 24, 42, 26, 50, 20, 54, 30, 49, 0, -8, -6, 4, 4, -4, -2, 11, 0, 16, 10, -7, 12, 15, 2, 26, 16, 13, 18, 35, 4, 37, 22, 18, 24, 47, 12

Offset: 1

Author

Antti Karttunen, Mar 12 2018

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384

Crossrefs

Cf. A005187, A008683, A053645, A297111, A300724, A300725.

Programs

Mathematica

With[{s = Array[2 # - DigitCount[2 #, 2, 1] &[# - 2^Floor@ Log2@ #] &, 90]}, Table[DivisorSum[n, MoebiusMu[n/#] s[[#]] &], {n, Length@ s}]] (* Michael De Vlieger, Mar 13 2018 *)

PARI

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A053644(n) = { my(k=1); while(k<=n, k<<=1); (k>>1); }; \\ From A053644
A053645(n) = (n-A053644(n));
A300723(n) = sumdiv(n,d,moebius(n/d)*A005187(A053645(d)));

Formula

a(n) = Sum_{d|n} A008683(n/d)*A005187(A053645(d)).

a(1) = 0; for n > 1, a(n) = A297111(n) - 2*A300724(n).