A376404 - OEIS

%I #9 Nov 15 2024 15:12:02

%S 1,0,0,2,-2,4,-2,10,8,4,-8,16,-8,8,8,42,-14,28,-14,12,16,4,-16,72,6,8,

%T 64,28,-26,16,-24,170,8,4,20,144,-32,8,16,52,-38,40,-38,0,40,12,-40,

%U 328,30,28,8,16,-46,228,24,124,16,4,-56,112,-54,12,96,682,32,-8,-62,-12,24,24,-68,712,-66,8,56,4,32,16

%N Dirichlet inverse of 2*phi(n) - phi(A003961(n)), where phi is Euler totient function and A003961(n) is fully multiplicative function with a(prime(i)) = prime(i+1).

%H Antti Karttunen, <a href="/A376404/b376404.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>

%F a(1) = 1, and for n > 1, a(n) = -Sum_{d|n, d<n} -A349754(n/d) * a(d).

%F a(n) = Sum_{d|n} A346248(d).

%o (PARI)

%o A003961(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); factorback(f); };

%o A349754(n) = (eulerphi(A003961(n))-2*eulerphi(n));

%o memoA376404 = Map();

%o A376404(n) = if(1==n,1,my(v); if(mapisdefined(memoA376404,n,&v), v, v = -sumdiv(n,d,if(d<n,-A349754(n/d)*A376404(d),0)); mapput(memoA376404,n,v); (v)));

%Y Cf. A000010, A003961, A003972.

%Y Dirichlet inverse of -A349754, inverse Möbius transform of A346248.

%Y Cf. also A323912.

%K sign

%O 1,4

%A _Antti Karttunen_, Nov 15 2024