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A297111 Möbius transform of A005187, where A005187(n) = 2n - (number of 1's in binary representation of n). 12

%I

%S 1,2,3,4,7,4,10,8,12,8,18,8,22,12,15,16,31,12,34,16,25,20,41,16,39,24,

%T 34,24,53,16,56,32,42,32,49,24,70,36,48,32,78,24,81,40,48,44,88,32,84,

%U 40,63,48,101,36,79,48,72,56,112,32,116,60,69,64,98,40,130,64,90,48,137,48,142,72,81,72,121,48,152,64

%N Möbius transform of A005187, where A005187(n) = 2n - (number of 1's in binary representation of n).

%C Sequence differs from A035532 for the first time at n = 15, 21, 25, 27, 33, 35, 51, etc., i.e., at those composite n where A297115 has a nonzero value. - _Antti Karttunen_ & _M. F. Hasler_, Mar 10 2018

%H Antti Karttunen, <a href="/A297111/b297111.txt">Table of n, a(n) for n = 1..16384</a>

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

%F a(n) = n + A297114(n).

%F From _Antti Karttunen_, Mar 11 2018: (Start)

%F Sum A005187(n) x^n = Sum a(n)*x^n/(1-x^n). [Another way of saying that this is the Möbius transform of A005187. This was originally included in A035532 by mistake.]

%F a(n) = 2*phi(n) - A297115(n) = phi(n) + A297117(n).

%F a(n) = A005187(n) - A300244(n).

%F a(1) = 1; for n > 1, a(n) = A300723(n) + 2*A300724(n).

%F (End)

%t Table[DivisorSum[n, IntegerExponent[(2 #)!, 2] MoebiusMu[n/#] &], {n, 80}] (* _Michael De Vlieger_, Mar 10 2018 *)

%o (PARI)

%o A005187(n) = { my(s=n); while(n>>=1, s+=n); s; };

%o A297111(n) = sumdiv(n,d,moebius(n/d)*A005187(d));

%Y Cf. A000010, A005187, A008683, A297108, A297110, A297114, A297115, A297117, A300244, A300723, A300724, A300725.

%K nonn

%O 1,2

%A _Antti Karttunen_, Dec 25 2017

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Last modified February 21 01:29 EST 2019. Contains 320364 sequences. (Running on oeis4.)