A228409 a(n) = 4*mu(n) + 5, where mu is the Moebius function (A008683).

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, 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, 9, 1, 1, 5, 1, 9, 5, 5, 9, 1, 1, 5, 5, 9, 1, 5, 9, 9, 9

Offset: 1

Author

Wesley Ivan Hurt, Nov 09 2013

Keywords

• nonn
• easy

Comments

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).

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.

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

Examples

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

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n

Crossrefs

Cf. A000040, A001358, A008683, A030059.

Programs

Maple

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

Mathematica

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

PARI

a(n)=4*moebius(n)+5 \\ Charles R Greathouse IV, Nov 12 2013

Scheme

(define (A228409 n) (+ 5 (* 4 (A008683 n)))) ;; Antti Karttunen, Jul 26 2017