A186970 The oex analog of the Euler phi-function for the oex prime power factorization of positive integers.

1, 1, 2, 3, 4, 3, 6, 4, 8, 5, 10, 7, 12, 8, 9, 12, 16, 11, 18, 12, 14, 14, 22, 9, 24, 16, 18, 18, 28, 13, 30, 16, 22, 21, 25, 24, 36, 24, 27, 17, 40, 17, 42, 30, 33, 29, 46, 27, 48, 32, 36, 36, 52, 24, 42, 25, 40, 37, 58, 28, 60, 40, 49, 48, 50, 30, 66, 48, 49, 35, 70, 32, 72, 48, 54, 54, 61, 36

Offset: 1

Author

Vladimir Shevelev, Mar 01 2011

Keywords

• nonn

Comments

Oex divisors d of an integer n are defined in A186443: those divisors d which are either 1 or numbers such that d^k || n (the highest power of d dividing n) has odd exponent k.

A positive number is called an oex prime if it has only two oex divisors; since every n >= 2 has at least two oex divisors, 1 and n, an oex prime q has only oex divisors 1 and q. A000430 is the sequence of oex primes q, i.e., A186643(q) = 2 iff q is an entry in A000430.

A unique factorization, called an oex prime power factorization, of integers n is introduced as follows: each factor p^e in the conventional prime power factorization n = Product(p^e) is written as (p^2)^(e/2) if e is even, and as (p^2)^floor(e/2)*p if e is odd. This represents n as a product of oex primes of the type q=p^2, with unconstrained exponents e/2, and of oex primes of the type q=p with exponents 0 or 1. (This is similar to splitting n into its squarefree part A007913(n) times A008833(n), followed by an ordinary prime factorization in both parts separately.)

Let n = q_1^a_1*q_2^a_2*... and m = q_1^b_1*q_2^b_2*..., a_i,b_i >= 0 be the oex prime power factorizations of n and m. Define the oex GCD of n and m as [n,m] = q_1^min(a_1,b_1) * q_2^min(a_2,b_2) * .... Then a(n) = Sum_{m=1..n, [m,n]=1} 1, the oex analog of the Euler-phi function.

Examples

			The oex prime power factorization of 16 is 4^2. Since [16,i]=1 for i=1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, and 15, a(16)=12.
The oex prime power factorization of 9 is 9. Thus a(9)=8.
		

Crossrefs

Cf. A000010, A000430, A186443, A186444, A186776, A064380.

Programs

Maple

highpp := proc(n,d) local nshf,a ; if n mod d <> 0 then 0; else nshf := n ; a := 0 ; while nshf mod d = 0 do nshf := nshf /d ; a := a+1 ; end do: a; end if; end proc:
oexgcd := proc(n,m) local a,p,kn,km ; a := 1 ; for p in numtheory[factorset](n) do kn := highpp(n,p) ; km := highpp(m,p) ; if type(kn,'even') = type(km,'even') then ; else kn := 2*floor(kn/2) ; km := 2*floor(km/2) ; end if; a := a*p^min(kn,km) ; end do: a ; end proc:
A186970 := proc(n) local a,i; a := 0 ; for i from 1 to n do if oexgcd(n,i) = 1 then a := a+1 ; end if; end do: a; end proc:
seq(A186970(n),n=1..80) ; # R. J. Mathar, Mar 18 2011

Formula

Let core(n) = p_1*...*p_r = A007913(n), n/core(n) = A008833(n) = q_1^c_1*...*q_t^c_t, where q_i are squares of primes.

If core(n)=1, then a(n) = n*Product_{j=1..r} (1-1/q_i); if core(n) tends to infinity, then a(n) ~ n * core(n) * Product_{i=1..t} (1-1/q_i) / Product_{j=1..r} (1+p_j).

a(n) <= A064380(n).