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%I #11 Jan 29 2019 23:59:59
%S 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,
%T 18,28,13,30,16,22,21,25,24,36,24,27,17,40,17,42,30,33,29,46,27,48,32,
%U 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
%N The oex analog of the Euler phi-function for the oex prime power factorization of positive integers.
%C 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.
%C 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.
%C 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.)
%C 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.
%F 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.
%F 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).
%F a(n) <= A064380(n).
%e 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.
%e The oex prime power factorization of 9 is 9. Thus a(9)=8.
%p 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:
%p 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:
%p 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:
%p seq(A186970(n),n=1..80) ; # _R. J. Mathar_, Mar 18 2011
%Y Cf. A000010, A000430, A186443, A186444, A186776, A064380.
%K nonn
%O 1,3
%A _Vladimir Shevelev_, Mar 01 2011
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