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%I #24 Mar 24 2018 18:53:21
%S 0,0,0,1,0,3,0,3,2,5,0,7,0,9,6,7,0,9,0,11,10,17,0,15,4,33,6,19,0,17,0,
%T 15,18,65,12,19,0,129,34,23,0,29,0,35,14,257,0,31,8,17,66,67,0,21,20,
%U 39,130,513,0,35,0,1025,22,31,36,53,0,131,258,33,0,39,0,2049,18,259,24,101,0,47,14,4097,0,59,68,8193,514,71,0,37,40
%N Difference between A156552 and its Moebius transform: a(n) = A156552(n) - A297112(n).
%H Antti Karttunen, <a href="/A297168/b297168.txt">Table of n, a(n) for n = 1..8192</a>
%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>
%F a(n) = -Sum_{d|n, d<n} A008683(n/d)*A156552(d).
%F a(n) = Sum_{d|n, d<n} A297112(d).
%F For n > 1, a(n) = Sum_{d|n, 1<d<n} 2^A033265(A156552(d)).
%F a(n) = A156552(n) - A297112(n).
%F a(1) = 0, for n > 1, a(n) = A156552(n) - 2^A297167(n).
%t With[{s = Array[Total@ MapIndexed[#1 2^(First@ #2 - 1) &, Flatten@ Map[ConstantArray[2^(PrimePi@ #1 - 1), #2] & @@ # &, FactorInteger@ #]] - Boole[# == 1]/2 &, 91]}, Table[-DivisorSum[n, MoebiusMu[n/#] s[[#]] &, # < n &], {n, Length@ s}]] (* _Michael De Vlieger_, Mar 13 2018 *)
%o (PARI)
%o 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 A156552(n) = if(1==n, 0, if(!(n%2), 1+(2*A156552(n/2)), 2*A156552(A064989(n))));
%o A297112(n) = sumdiv(n,d,moebius(n/d)*A156552(d));
%o A297168(n) = (A156552(n)-A297112(n));
%o \\ Or alternatively as:
%o A297168(n) = -sumdiv(n,d,(d<n)*moebius(n/d)*A156552(d));
%o (PARI)
%o A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
%o A297167(n) = if(1==n, 0, (A061395(n) + (bigomega(n)-omega(n)) - 1));
%o A297112(n) = if(1==n,0,2^A297167(n));
%o A297168(n) = sumdiv(n,d,(d<n)*A297112(d)); \\ _Antti Karttunen_, Mar 13 2018
%o (Scheme)
%o (define (A297168 n) (- (A156552 n) (A297112 n)))
%o (define (A297168 n) (if (= 1 n) 0 (- (A156552 n) (A000079 (A297167 n)))))
%Y Cf. A008683, A033265, A156552, A297112, A297112, A297113, A297167, A297169, A300827.
%K nonn
%O 1,6
%A _Antti Karttunen_, Feb 27 2018
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