A300244 Difference between A005187 and its Möbius transform (A297111).

0, 1, 1, 3, 1, 6, 1, 7, 4, 10, 1, 14, 1, 13, 11, 15, 1, 22, 1, 22, 14, 21, 1, 30, 8, 25, 16, 29, 1, 40, 1, 31, 22, 34, 18, 46, 1, 37, 26, 46, 1, 57, 1, 45, 38, 44, 1, 62, 11, 57, 35, 53, 1, 68, 26, 61, 38, 56, 1, 84, 1, 59, 51, 63, 30, 90, 1, 70, 45, 89, 1, 94, 1, 73, 65, 77, 29, 104, 1, 94, 50, 81, 1, 117, 39, 84, 57, 93, 1, 128, 33, 92, 60, 91, 42

Offset: 1

Author

Antti Karttunen, Mar 10 2018

Keywords

• nonn

Links

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

Crossrefs

Cf. A005187, A008683, A297111, A297114, A297117.

Programs

Mathematica

Table[IntegerExponent[(2 n)!, 2] - DivisorSum[n, IntegerExponent[(2 #)!, 2] MoebiusMu[n/#] &], {n, 95}] (* or *)
Fold[Function[{a, n}, Append[a, {Abs@ Total@ Map[MoebiusMu[n/#] a[[#, -1]] &, Most@ Divisors@ n], IntegerExponent[(2 n)!, 2]}]], {{0, 1}}, Range[2, 95]][[All, 1]] (* Michael De Vlieger, Mar 10 2018 *)

PARI

A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };
A300244(n) = -sumdiv(n,d,(dA005187(d));

Formula

a(n) = A005187(n) - A297111(n).

a(n) = -Sum_{d|n, dA008683(n/d)*A005187(d).