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Dirichlet convolution of A001221 (omega) with A003602 (Kimberling's paraphrases).
5

%I #21 Nov 18 2021 09:02:32

%S 0,1,1,2,1,5,1,3,3,6,1,9,1,7,7,4,1,14,1,11,8,9,1,13,4,10,8,13,1,28,1,

%T 5,10,12,9,25,1,13,11,16,1,34,1,17,22,15,1,17,5,25,13,19,1,38,11,19,

%U 14,18,1,49,1,19,26,6,12,46,1,23,16,44,1,36,1,22,31,25,12,52,1,21,22,24,1,60,14,25,19,25,1,86

%N Dirichlet convolution of A001221 (omega) with A003602 (Kimberling's paraphrases).

%H Antti Karttunen, <a href="/A347957/b347957.txt">Table of n, a(n) for n = 1..16384</a>

%H Jon Maiga, <a href="http://sequencedb.net/s/A347957">Computer-generated formulas for A347957</a>, Sequence Machine.

%F a(n) = Sum_{d|n} A001221(n/d) * A003602(d).

%F From _Antti Karttunen_, Nov 13 2021: (Start)

%F The following two convolutions were found by Jon Maiga's Sequence Machine search algorithm. The first one is obvious, and even the second one should not be too hard to prove:

%F a(n) = Sum_{d|n} A023900(n/d) * A347956(d).

%F a(n) = Sum_{d|n} A181988(n/d) * A205745(d).

%F (End)

%o (PARI)

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

%o A347957(n) = sumdiv(n,d,omega(n/d)*A003602(d));

%Y Cf. A001221, A003602, A023900, A181988, A205745.

%Y Cf. also A328203, A347954, A347955, A347956.

%K nonn

%O 1,4

%A _Antti Karttunen_, Sep 20 2021