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A263774 Sequence is defined by the condition that Sum_{d|n} a(d)^(n/d) = mu(n)^2, where mu(n) is the Möbius function.

%I A263774 #45 Mar 27 2017 10:25:16
%S A263774 1,0,0,-1,0,0,0,-2,-1,0,0,0,0,0,0,-6,0,-2,0,0,0,0,0,6,-1,0,0,0,0,0,0,
%T A263774 -54,0,0,0,-5,0,0,0,30,0,0,0,0,0,0,0,114,-1,-2,0,0,0,6,0,126,0,0,0,0,
%U A263774 0,0,0,-4470,0,0,0,0,0,0,0,252,0,0,0,0,0,0,0
%N A263774 Sequence is defined by the condition that Sum_{d|n} a(d)^(n/d) = mu(n)^2, where mu(n) is the Möbius function.
%H A263774 Reinhard Zumkeller, <a href="/A263774/b263774.txt">Table of n, a(n) for n = 1..10000</a>
%F A263774 a(1) = 1.
%F A263774 If p>2, a(p) = 0, a(p^2) = -1, a(p^n) = 0 for n>2.
%F A263774 a(p1*p2*..*pn) = 0, a(2*p1*...*pn) = 0, a(4*p1*...*pn) = 0.
%F A263774 If p1,...,pn are odd it appears that:
%F A263774 a(p1^2*p2^2*...*pn^2) = (-1)^n,
%F A263774 a(p1^k1*p2^k2*...*pn^kn) = 0, if one of k1,...,kn > 2,
%F A263774 a(2*p1^k1*p2^k2*...*pn^kn) > 0 if one of k1,...,kn > 1.
%F A263774 a(2^n) = A264610(n).
%e A263774 For a prime p, a(p)^1 + a(1)^p = mu(p)^2=1 => a(p) = 0.
%e A263774 For n=6, a(1)^6 + a(2)^3 + a(3)^2 + a(6)^1 = mu(6) = 1, so 1 - 0 + 0 + a(6) = 1, so 1 + a(6) = 1, so a(6) = 0.
%p A263774 a := proc (n) option remember; numtheory:-mobius(n)^2-add(procname(n/d)^d, d = `minus`(numtheory:-divisors(n), {1})) end proc; a(1) := 1; La := seq(a(i), i = 1 .. 100)
%t A263774 a[n_]:=If[n<2, 1, MoebiusMu[n]^2 - Sum[If[d==1, 0, a[n/d]^d], {d, Divisors[n]}]]; Table[a[n], {n, 100}] (* _Indranil Ghosh_, Mar 26 2017 *)
%o A263774 (PARI) a(n) = if (n==1, 1, moebius(n)^2- sumdiv(n, d, if (d==1, 0, a(n/d)^d)));
%o A263774 (Haskell)
%o A263774 a263774 1 = 1
%o A263774 a263774 n = foldl (-) (a008966 n) $ zipWith (^) (map a' $ reverse ds) ds
%o A263774             where a' x = if x == n then 0 else a263774 x
%o A263774                   ds = a027750_row n
%o A263774 -- _Reinhard Zumkeller_, Dec 06 2015
%Y A263774 Cf. A008683, A264610.
%Y A263774 Cf. A008966, A027750.
%K A263774 sign
%O A263774 1,8
%A A263774 _Gevorg Hmayakyan_, Nov 28 2015