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Inverse Möbius transform of Kimberling's paraphrases (A003602).
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%I #17 Nov 21 2021 10:16:58

%S 1,2,3,3,4,6,5,4,8,8,7,9,8,10,14,5,10,16,11,12,18,14,13,12,17,16,22,

%T 15,16,28,17,6,26,20,26,24,20,22,30,16,22,36,23,21,42,26,25,15,30,34,

%U 38,24,28,44,38,20,42,32,31,42,32,34,55,7,44,52,35,30,50,52,37,32,38,40,65,33,50,60,41,20,63,44,43

%N Inverse Möbius transform of Kimberling's paraphrases (A003602).

%C Dirichlet convolution of sigma (A000203) with A349431, or equally, A264740 with A349447. - _Antti Karttunen_, Nov 21 2021

%H Antti Karttunen, <a href="/A349371/b349371.txt">Table of n, a(n) for n = 1..20000</a>

%F a(n) = Sum_{d|n} A003602(d).

%F a(n) = Sum_{d|n} A000203(n/d)*A349431(d) = Sum_{d|n} A264740(n/d)*A349447(d). - _Antti Karttunen_, Nov 21 2021

%t k[n_] := (n / 2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, k[#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 16 2021 *)

%o (PARI)

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

%o A349371(n) = sumdiv(n,d,A003602(d));

%Y Cf. A000203, A264740.

%Y Cf. also A347954, A347955, A347956, A349136, A349370, A349372, A349373, A349374, A349375, A349390, A349431, A349444, A349447 for Dirichlet convolutions of other sequences with A003602.

%Y Cf. also A349393.

%K nonn

%O 1,2

%A _Antti Karttunen_, Nov 15 2021