A355690 - OEIS

%I #20 Jan 12 2023 18:43:19

%S 1,0,-1,-1,-1,0,-1,-1,0,0,-1,1,-1,0,1,0,-1,0,-1,1,1,0,-1,1,0,0,0,1,-1,

%T 0,-1,1,1,0,1,0,-1,0,1,1,-1,0,-1,1,0,0,-1,0,0,0,1,1,-1,0,1,1,1,0,-1,

%U -1,-1,0,0,1,1,0,-1,1,1,0,-1,0,-1,0,0,1,1,0,-1,0,0,0,-1,-1,1,0,1,1,-1,0,1,1,1,0,1,-1,-1,0,0,0,-1,0,-1,1,-1,0,-1,0

%N Dirichlet inverse of A152822, where A152822 is the characteristic function of numbers not congruent to 2 mod 4.

%C Multiplicative because A152822 is.

%H Antti Karttunen, <a href="/A355690/b355690.txt">Table of n, a(n) for n = 1..100000</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A152822(n/d) * a(d).

%F Multiplicative with a(2^e) = A010892(1+e), and for odd primes p, a(p^e) = -1 if e = 1, otherwise 0. - _Antti Karttunen_, Dec 23 2022

%F a(n) = A359605(n) - A359606(n). - _Antti Karttunen_, Jan 12 2023

%o (PARI)

%o A152822(n) = (2!=(n%4));

%o memoA355690 = Map();

%o A355690(n) = if(1==n,1,my(v); if(mapisdefined(memoA355690,n,&v), v, v = -sumdiv(n,d,if(d<n,A152822(n/d)*A355690(d),0)); mapput(memoA355690,n,v); (v)));

%o (PARI)

%o A010892(n) = ([1, 1, 0, -1, -1, 0][n%6 + 1]);

%o A355690(n) = { my(f = factor(n)); prod(k=1, #f~, if(2==f[k,1], A010892(1+f[k, 2]), -(1==f[k, 2]))); }; \\ _Antti Karttunen_, Dec 23 2022

%Y Cf. A010892, A042965, A152822, A359590 (absolute values), A359605, A359606.

%Y Cf. also A355688, A355689, A355691.

%K sign,mult,easy

%O 1

%A _Antti Karttunen_, Jul 15 2022