A355936 - OEIS

%I #15 Feb 27 2023 12:18:46

%S 1,-1,-1,1,-1,1,-1,-2,1,1,-1,-1,-1,1,1,3,-1,-1,-1,-1,1,1,-1,2,1,1,-2,

%T -1,-1,-1,-1,-5,1,1,1,1,-1,1,1,2,-1,-1,-1,-1,-1,1,-1,-3,1,-1,1,-1,-1,

%U 2,1,2,1,1,-1,1,-1,1,-1,8,1,-1,-1,-1,1,-1,-1,-2,-1,1,-1,-1,1,-1,-1,-3,3,1,-1,1,1,1,1,2,-1,1,1,-1,1,1,1,5,-1,-1,-1,1,-1,-1,-1,2,-1

%N Dirichlet inverse of A295316, characteristic function of exponentially odd numbers.

%C Multiplicative because A295316 is.

%H Antti Karttunen, <a href="/A355936/b355936.txt">Table of n, a(n) for n = 1..100000</a>

%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A295316(n/d) * a(d).

%F Multiplicative with a(p^e) = (-1)^e * Fibonacci(e). - _Sebastian Karlsson_, Jul 24 2022

%F Sum_{k=1..n} abs(a(k)) ~ c * n, where c = Product_{primes p} (1 + 1/(p^3 - p^2 - p)) = 1.6256655992867552241340804110236555506570411887342367924818823782775... - _Vaclav Kotesovec_, Feb 27 2023

%t s[n_] := If[AllTrue[FactorInteger[n][[;; , -1]], OddQ], 1, 0]; a[1] = 1; a[n_] := -DivisorSum[n, a[#]*s[n/#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Jul 21 2022 *)

%o (PARI)

%o A295316(n) = factorback(apply(e -> (e%2), factorint(n)[, 2]));

%o memoA355936 = Map();

%o A355936(n) = if(1==n,1,my(v); if(mapisdefined(memoA355936,n,&v), v, v = -sumdiv(n,d,if(d<n,A295316(n/d)*A355936(d),0)); mapput(memoA355936,n,v); (v)));

%Y Cf. A268335, A295316.

%Y Cf. also A355826.

%Y Cf. A000045.

%K sign,mult

%O 1,8

%A _Antti Karttunen_, Jul 21 2022