A353459 - OEIS

%I #18 Jan 05 2023 11:22:29

%S 2,0,0,1,0,-2,0,1,1,2,0,-1,0,-2,-2,1,0,-1,0,1,2,2,0,-1,1,-2,1,-1,0,0,

%T 0,1,-2,2,-2,0,0,-2,2,1,0,0,0,1,-1,2,0,-1,1,1,-2,-1,0,-1,2,-1,2,-2,0,

%U -1,0,2,1,1,-2,0,0,1,-2,0,0,0,0,-2,-1,-1,-2,0,0,1,1,2,0,1,2,-2,2,1,0,1,2,1,-2,2,-2,-1,0

%N Sum of A353457 and its Dirichlet inverse.

%C Only values in range {-2, -1, 0, +1, +2} occur.

%H Antti Karttunen, <a href="/A353459/b353459.txt">Table of n, a(n) for n = 1..65537</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>

%F a(n) = A353457(n) + A353458(n) = A353457(n) + A353457(A064989(n)).

%F For n > 1, a(n) = -Sum_{d|n, 1<d<n} A353457(d) * A353458(n/d). [As the sequences are Dirichlet inverses of each other]

%F For all n >= 1, a(n) = a(A003961(n)) = a(A348717(n)).

%o (PARI)

%o A000265(n) = (n>>valuation(n,2));

%o A064989(n) = { my(f=factor(A000265(n))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };

%o memoA353457 = Map();

%o A353457(n) = if(1==n,1,my(v); if(mapisdefined(memoA353457,n,&v), v, v = -sumdiv(n,d,if(d<n,A353457(A064989(n/d))*A353457(d),0)); mapput(memoA353457,n,v); (v)));

%o A353459(n) = (A353457(n)+A353457(A064989(n)));

%o (Python)

%o from math import prod

%o from sympy import factorint, primepi

%o def A353459(n):

%o     f = [(primepi(p)&1, -int(e==1)) for p, e in factorint(n).items()]

%o     return prod(e for p, e in f if not p)+prod(e for p, e in f if p) # _Chai Wah Wu_, Jan 05 2023

%Y Cf. A003961, A348717, A353457, A353458.

%Y Cf. also A353469.

%K sign

%O 1,1

%A _Antti Karttunen_, Apr 21 2022