A359164 - OEIS

%I #14 Dec 23 2022 11:25:19

%S 0,1,1,1,1,2,1,1,2,3,1,2,1,4,4,1,1,5,1,3,5,6,1,2,3,7,5,4,1,8,1,1,7,9,

%T 6,5,1,10,8,3,1,11,1,6,11,12,1,2,4,13,10,7,1,14,8,4,11,15,1,8,1,16,14,

%U 1,9,17,1,9,13,18,1,5,1,19,18,10,9,20,1,3,14,21,1,11,11,22,16,6,1,23,10,12,17,24,12

%N Difference between Kimberling's paraphrases and its Möbius transform.

%H Antti Karttunen, <a href="/A359164/b359164.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A359164/a359164.txt">Data supplement: n, a(n) computed for n = 1..65539</a>

%F a(n) = A003602(n) - A349136(n).

%F a(n) = Sum_{d|n, d<n} A349136(d).

%F a(n) = -Sum_{d|n, d<n} A008683(n/d)*A003602(d).

%F a((2^e)*prime(n)) = A111333(n) for all n >= 1 and e >= 1.

%o (PARI)

%o A003602(n) = (1+(n>>valuation(n,2)))/2;

%o A349136(n) = sumdiv(n,d,moebius(d)*A003602(n/d));

%o A359164(n) = (A003602(n) - A349136(n));

%Y Cf. A003602, A008683, A111333, A349136.

%Y Cf. also A349135, A359165.

%K nonn

%O 1,6

%A _Antti Karttunen_, Dec 22 2022