cp's OEIS Frontend

A366889 Dirichlet inverse of the highest power of two that divides the sum of divisors of n.

%I A366889 #9 Jan 03 2024 14:37:36
%S A366889 1,-1,-4,0,-2,4,-8,0,15,2,-4,0,-2,8,8,0,-2,-15,-4,0,32,4,-8,0,3,2,-64,
%T A366889 0,-2,-8,-32,0,16,2,16,0,-2,4,8,0,-2,-32,-4,0,-30,8,-16,0,63,-3,8,0,
%U A366889 -2,64,8,0,16,2,-4,0,-2,32,-120,0,4,-16,-4,0,32,-16,-8,0,-2,2,-12,0,32,-8,-16,0,272,2,-4,0,4,4,8,0
%N A366889 Dirichlet inverse of the highest power of two that divides the sum of divisors of n.
%C A366889 Multiplicative because A082903 is.
%H A366889 Antti Karttunen, <a href="/A366889/b366889.txt">Table of n, a(n) for n = 1..16384</a>
%H A366889 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>
%F A366889 a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A082903(n/d) * a(d).
%o A366889 (PARI)
%o A366889 A082903(n) = (2^valuation(sigma(n), 2));
%o A366889 memoA366889 = Map();
%o A366889 A366889(n) = if(1==n,1,my(v); if(mapisdefined(memoA366889,n,&v), v, v = -sumdiv(n,d,if(d<n,A082903(n/d)*A366889(d),0)); mapput(memoA366889,n,v); (v)));
%o A366889 (Python)
%o A366889 from functools import lru_cache
%o A366889 from sympy import divisor_sigma, divisors
%o A366889 @lru_cache(maxsize=None)
%o A366889 def A366889(n): return 1 if n==1 else -sum((1<<(~(m:=int(divisor_sigma(d))) & m-1).bit_length())*A366889(n//d) for d in divisors(n,generator=True) if d>1) # _Chai Wah Wu_, Jan 03 2024
%Y A366889 Cf. A000203, A082903, A336937, A359548, A359549 (parity of terms).
%K A366889 sign,mult
%O A366889 1,3
%A A366889 _Antti Karttunen_, Jan 03 2024