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%I #16 Jul 23 2017 21:46:04
%S 0,-1,1,1,1,1,0,0,1,-1,-1,1,-1,-1,1,0,1,1,-1,-1,1,-1,1,0,0,-1,-1,1,-1,
%T -1,0,-1,-1,-1,0,0,1,0,1,-1,1,-1,-1,1,0,0,1,-1,1,-1,0,-1,0,0,-1,1,0,0,
%U 1,-1,-1,0,0,-1,1,-1,1,0,1,0,-1,1,-1,-1,0,1,1,1,-1,-1,-1,1,-1,-1,-1,-1,0,1,1,1,1,1,0,0,-1,1,1,0,0,1,1,0,1,1,1,0,1,0,-1
%N a(n) = mu(prime(n)+1) + mu(prime(n)-1), where mu is the Moebius function.
%C This sum is always -1, 0 or 1 because for odd prime p, both p-1 and p+1 cannot be squarefree; one of them will be divisible by 4. This also implies that terms in this sequence are zero only for 2 and odd primes p such that mu(p-1) = mu(p+1) = 0, which is A075432.
%H Antti Karttunen, <a href="/A089496/b089496.txt">Table of n, a(n) for n = 1..10000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MoebiusFunction.html">Moebius Function</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LegendreSymbol.html">Legendre Symbol</a>
%F Let p = prime(n), then a(n) = mu(p+(-1/p)), where (-1/p) is the Legendre symbol, A070750. (Pieter Moree). (This is true for n > 1) - _Antti Karttunen_, Jul 23 2017
%t Table[MoebiusMu[Prime[n]+1] + MoebiusMu[Prime[n]-1], {n, 1, 150}]
%o (PARI) A089496(n) = (moebius(prime(n)-1)+moebius(prime(n)+1)); \\ _Antti Karttunen_, Jul 23 2017
%Y Cf. A000040, A008683, A089451 (mu(p-1) for prime p), A089495 (mu(p+1) for prime p), A089497 (mu(p+1)-mu(p-1) for prime p).
%K sign
%O 1
%A _T. D. Noe_, Nov 04 2003
%E Term a(1) = 0 prepended by _Antti Karttunen_, Jul 23 2017
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