A297114 Möbius transform of A294898, where A294898 is deficiency minus binary weight.

0, 0, 0, 0, 2, -2, 3, 0, 3, -2, 7, -4, 9, -2, 0, 0, 14, -6, 15, -4, 4, -2, 18, -8, 14, -2, 7, -4, 24, -14, 25, 0, 9, -2, 14, -12, 33, -2, 9, -8, 37, -18, 38, -4, 3, -2, 41, -16, 35, -10, 12, -4, 48, -18, 24, -8, 15, -2, 53, -28, 55, -2, 6, 0, 33, -26, 63, -4, 21, -22, 66, -24, 69, -2, 6, -4, 44, -30, 73, -16, 28, -2

Offset: 1

Author

Antti Karttunen, Dec 26 2017

Keywords

• sign

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to binary expansion of n
• Index entries for sequences related to sigma(n)

Crossrefs

Cf. A000203, A005187, A051953, A083254, A294898, A297111, A297115, A297117, A317844, A324397.

Cf. A378986 (even bisection, -2*A176095).

Programs

Mathematica

Table[DivisorSum[n, MoebiusMu[n/#] (2 # - DigitCount[2 #, 2, 1] - DivisorSigma[1, #]) &], {n, 82}] (* Michael De Vlieger, Mar 11 2019 *)

PARI

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A297114(n) = sumdiv(n,d,moebius(n/d)*(A005187(d)-sigma(d)));

Formula

a(n) = Sum_{d|n} A008683(n/d)*A294898(d).

a(n) = A297111(n) - n.

a(n) = A297117(n) - A051953(n).

a(n) = A083254(n) - A297115(n).

a(2n) = A083254(2n) = A378986(n) = -2*A176095(n).

a(n) = A294898(n) - A317844(n).