A355827 - OEIS

%I #16 Jul 20 2022 08:50:34

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

%T -1,-1,-6,-1,2,2,2,2,2,-1,2,2,-2,-1,-6,-1,-1,-1,2,-1,6,0,-1,2,-1,-1,

%U -2,2,-2,2,2,-1,6,-1,2,-1,0,2,-6,-1,-1,2,-6,-1,0,-1,2,-1,-1,2,-6,-1,6,-2,2,-1,6,2,2,2,-2,-1,6,2

%N a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} A302777(n/d) * a(d).

%C Dirichlet inverse of function f(1) = 1, f(n) = A302777(n) for n > 1, which is the characteristic function of the union of {1} and "Fermi-Dirac primes", A050376.

%H Antti Karttunen, <a href="/A355827/b355827.txt">Table of n, a(n) for n = 1..16384</a>

%H Antti Karttunen, <a href="/A355827/a355827.txt">Data supplement: n, a(n) computed for n = 1..100000</a>

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

%t s[n_] := If[n > 1 && Length[(f = FactorInteger[n])] == 1 && (e = f[[;; , 2]]) == 2^IntegerExponent[e, 2], 1, 0]; a[1] = 1; a[n_] := a[n] = -DivisorSum[n, s[n/#] * a[#] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Jul 19 2022 *)

%o (PARI)

%o ispow2(n) = (n && !bitand(n,n-1));

%o A302777(n) = ispow2(isprimepower(n));

%o memoA355827 = Map();

%o A355827(n) = if(1==n,1,my(v); if(mapisdefined(memoA355827,n,&v), v, v = -sumdiv(n,d,if(d<n,A302777(n/d)*A355827(d),0)); mapput(memoA355827,n,v); (v)));

%Y Cf. A050376, A302777.

%Y Cf. also A355817.

%K sign

%O 1,6

%A _Antti Karttunen_, Jul 19 2022