A228409 - OEIS

%I #32 Oct 03 2017 03:20:02

%S 9,1,1,5,1,9,1,5,5,9,1,5,1,9,9,5,1,5,1,5,9,9,1,5,5,9,5,5,1,1,1,5,9,9,

%T 9,5,1,9,9,5,1,1,1,5,5,9,1,5,5,5,9,5,1,5,9,5,9,9,1,5,1,9,5,5,9,1,1,5,

%U 9,1,1,5,1,9,5,5,9,1,1,5,5,9,1,5,9,9,9

%N a(n) = 4*mu(n) + 5, where mu is the Moebius function (A008683).

%C If n is prime (A000040), then a(n) = 1. The converse is not true: when n is the product of an odd number of distinct primes, mu(n) = -1 => a(n) = 1 (30 = 2*3*5, so a(30) = 1).

%C If n is semiprime (A001358), a(n) gives the number of divisors of n^2. In particular, if n = p^2 then n^2 = (p^2)^2 = p^4 has 5 divisors: p^4, p^3, p^2, p, 1. If n = pq (p,q distinct primes) then n^2 = (pq)^2 has 9 divisors: (pq)^2, qp^2, pq^2, p^2, q^2, pq, p, q, and 1.

%C a(n) = 1 if and only if n has an odd number of distinct prime factors, A030059. - _Jon Perry_, Nov 12 2013.

%H Antti Karttunen, <a href="/A228409/b228409.txt">Table of n, a(n) for n = 1..10000</a>

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

%e a(6) = 9; 4*mu(6) + 5 = 4*1 + 5 = 9.

%p with(numtheory); A228409:=n->4*mobius(n)+5; seq(A228409(n), n=1..100);

%t Table[4 MoebiusMu[n] + 5, {n, 100}]

%o (PARI) a(n)=4*moebius(n)+5 \\ _Charles R Greathouse IV_, Nov 12 2013

%o (Scheme) (define (A228409 n) (+ 5 (* 4 (A008683 n)))) ;; _Antti Karttunen_, Jul 26 2017

%Y Cf. A000040, A001358, A008683, A030059.

%K nonn,easy

%O 1,1

%A _Wesley Ivan Hurt_, Nov 09 2013