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Möbius transform of A344695, gcd(sigma(n), psi(n)).
3

%I #17 Aug 26 2021 16:07:03

%S 1,2,3,-2,5,6,7,2,-3,10,11,-6,13,14,15,-2,17,-6,19,-10,21,22,23,6,-5,

%T 26,3,-14,29,30,31,2,33,34,35,6,37,38,39,10,41,42,43,-22,-15,46,47,-6,

%U -7,-10,51,-26,53,6,55,14,57,58,59,-30,61,62,-21,-2,65,66,67,-34,69,70,71,-6,73,74,-15,-38,77,78,79

%N Möbius transform of A344695, gcd(sigma(n), psi(n)).

%C Not multiplicative because A344695 isn't either. For example, a(4) = -2, a(27) = 3, but a(108) = -2 != -6.

%C The absolute values are not equal to A007947. The first n where abs(a(n)) != A007947(n) is at n=108, with a(108) = -2, while A007947(108) = 6.

%C The first n such that a(n) does not divide n are: 196, 216, 392, 432, 441, 588, etc.

%C The zeros occur at n = 288, 576, 1440, 2016, 2880, 3168, 3744, 4032, etc.

%H Antti Karttunen, <a href="/A347230/b347230.txt">Table of n, a(n) for n = 1..16384</a>

%F a(n) = Sum_{d|n} A008683(n/d) * A344695(d).

%F a(n) = A344695(n) - A347231(n).

%o (PARI)

%o A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1))); \\ After code in A001615

%o A344695(n) = gcd(sigma(n), A001615(n));

%o A347230(n) = sumdiv(n,d,moebius(n/d)*A344695(d));

%Y Cf. A000203, A001615, A008683, A344695, A347228, A347231.

%Y Cf. also A007947, A063659.

%K sign

%O 1,2

%A _Antti Karttunen_, Aug 25 2021