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A297112 Möbius transform of A156552.

%I A297112 #26 Jan 17 2025 19:22:31
%S A297112 0,1,2,2,4,2,8,4,4,4,16,4,32,8,4,8,64,4,128,8,8,16,256,8,8,32,8,16,
%T A297112 512,4,1024,16,16,64,8,8,2048,128,32,16,4096,8,8192,32,8,256,16384,16,
%U A297112 16,8,64,64,32768,8,16,32,128,512,65536,8,131072,1024,16,32,32,16,262144,128,256,8,524288,16,1048576,2048,8
%N A297112 Möbius transform of A156552.
%H A297112 Antti Karttunen, <a href="/A297112/b297112.txt">Table of n, a(n) for n = 1..2048</a>
%F A297112 a(1) = 0, a(2) = 1, after which, a(2n+1) = 2*a(A064989(2n+1)), a(4n) = 2*a(2n), a(4n+2) = a(2n+1).
%F A297112 a(n) = Sum_{d|n} A008683(n/d)*A156552(d).
%F A297112 For n > 1, a(n) = A000079(A297113(n)-1).
%o A297112 (PARI)
%o A297112 A064989(n) = {my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f)};
%o A297112 A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
%o A297112 A297112(n) = sumdiv(n,d,moebius(n/d)*A156552(d));
%o A297112 (Scheme)
%o A297112 ;; With memoization-macro definec.
%o A297112 (definec (A297112 n) (cond ((<= n 2) (- n 1)) ((odd? n) (* 2 (A297112 (A064989 n)))) ((= 2 (modulo n 4)) (A297112 (/ n 2))) (else (* 2 (A297112 (/ n 2)))))) ;; _Antti Karttunen_, Dec 27 2017
%Y A297112 Cf. A008683, A064989, A156552, A297113.
%Y A297112 Cf. also A297106, A297156, A297171, A297172.
%K A297112 nonn
%O A297112 1,3
%A A297112 _Antti Karttunen_, Dec 26 2017