A349789 - OEIS

%I #13 Dec 06 2021 03:13:09

%S 1,-2,-3,2,-5,9,-7,-2,6,15,-11,-15,-13,21,25,2,-17,-27,-19,-25,35,33,

%T -23,21,20,39,-12,-35,-29,-105,-31,-2,55,51,63,66,-37,57,65,35,-41,

%U -147,-43,-55,-80,69,-47,-27,42,-85,85,-65,-53,72,99,49,95,87,-59,245

%N a(1) = 1; a(n) = -Sum_{d|n, d < n} gpf(n/d) * a(d).

%C Dirichlet inverse of A006530.

%H Antti Karttunen, <a href="/A349789/b349789.txt">Table of n, a(n) for n = 1..20000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GreatestPrimeFactor.html">Greatest Prime Factor</a>

%F From _Bernard Schott_, Dec 05 2021: (Start)

%F a(n) = -n iff n is prime.

%F a(2^k) = (-1)^k * 2 for k > 0. (End)

%t a[1] = 1; a[n_] := a[n] = -Sum[(FactorInteger[n/d][[-1, 1]]) a[d], {d, Most @ Divisors[n]}]; Table[a[n], {n, 1, 60}]

%o (PARI)

%o A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);

%o memoA349789 = Map();

%o A349789(n) = if(1==n,1,my(v); if(mapisdefined(memoA349789,n,&v), v, v = -sumdiv(n,d,if(d<n,A006530(n/d)*A349789(d),0)); mapput(memoA349789,n,v); (v))); \\ _Antti Karttunen_, Dec 05 2021

%Y Cf. A006530, A349790.

%K sign

%O 1,2

%A _Ilya Gutkovskiy_, Nov 30 2021