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The Sagher map sends Product p_i^e_i / Product q_i^f_i (p_i and q_i being distinct primes) to Product p_i^(2e_i) * Product q_i^(2f_i-1). This map is multiplicative.

A version of a greedy construction of an integer-valued function a such that a(n)/a(n+1) runs through all nonnegative rationals exactly once.

After initial 0, odd-indexed terms are integers in order with m repeated phi(m) times; even-indexed terms are the corresponding numbers <= m and relatively prime to m, in descending order. - Franklin T. Adams-Watters, Dec 06 2006

Partial sums of A116085, which is more elementary to compute, cf. examples. Sequence A154888 has an equivalent definition except that i=j is allowed there, which yields the one-term sum 1/1 as an additional possibility, and thus A154888(n) = a(n)+1. Sequence A115855 is also about the same problem but does not require the fractions to be distinct. - M. F. Hasler, Jul 14 2016

The reduced residue system modulo n used here is the set of numbers k from the set {0,1,...,n-1} which satisfy gcd(k,n)=1. There are phi(n) = A000010(n) such numbers k. Cf. A038566. See also the Apostol reference p. 133, and the Wikipedia link.

This is the m=3 member of a family of sequences, call them rmnS(m) (reduced mod n sum), with entries rmnS(m;n):=sum(binomial(k+m-1,m),0<=k<=n-1 with gcd(k,n)=1), m>=0, n>=1. Recall gcd(0,n)=n.

The members for m=0, 1, and 2 are A000010(n), A023896(n) and A127415(n), respectively, where in the last two the offset for n=1 should be taken as 0 (not 1).

The length of row n is given by A279400(n)

For the restricted residue systems modulo n see A038566. For the primes of A038566 (for n >= 3) see A112484.

The primes of the restricted residue system modulo the (composite) positive numbers without a primitive root, given in A033949, are of interest for the determination of the Dirichlet characters modulo the A033949 numbers. For prime numbers (A000040) or for composite positive numbers that have prime primitive roots (A279398) the Dirichlet characters are determined from those of the prime primitive root.

The totatives of n are the numbers k <= n with gcd(k,n) = 1.

A number k is strongly prime to n if and only if k <= n is prime to n and k does not divide n-1. See the link to 'Strong Coprimality'. (Our terminology follows the plea of Knuth, Graham and Patashnik in Concrete Mathematics, p. 115.)

The length of row n is A037225(n), for n >= 0.

The doubling sequence modulo N = 2*n+1, for n >= 0, has entries DS(N, s(N,i), j) = s(N,i)*2^j (mod N), with j >= 0, and certain positive odd integer seeds s(N, i), for i = 1, 2, ..., S(N) = A037226((N-1)/2), where gcd(s(N, i), N) = 1 (restricted seeds modulo N). These doubling sequences are periodic with period length P(N) = A002326((N-1)/2) (order of 2 modulo N). Only the periods (cycles) {DS(N, s(N, i), j)}_{j=0..P(N)-1}, for i = 1, 2, ..., S(N), are listed.

N = 1 (n=0) is special: one takes here the restricted residue system modulo N not as [0] but as [1]. The order of 2 modulo 1 is 1, because 2^1 == 1 (mod 1) (== 0 (mod 1)).

In order to obtain the complete system of doubling sequences one starts with seed s(N, 1) = 1, and if all numbers from the smallest positive reduced residue system modulo N (called RRS(N), given in row N of A038566) are obtained, i.e., if P(N) = #RRS(N) = phi(N) = A000010(N), then the system is complete. Otherwise the smallest missing number from RRS(N) is taken as new seed s(N, 2), etc. until the system is complete. This means that the number of seeds needed is S(N) = phi(N)/P(N) = A037226((N-1)/2)).

The irregular subtriangle where only seed s(N, 1) = 1 has been used is given in A201908. But there 0 (not 1) for N = 1 has been used.

From Gary W. Adamson and Wolfdieter Lang, Dec 15 2020: (Start)

The cycles in row n, for N = 2*n + 1, of period length P(N) = A002326((N-1)/2) give the periods of the iterated doubling function D(x) = frac(2*x) with seeds x = s(N, i)/N, for i = 1, 2, ..., S(N) = A037226((N-1)/2), after multiplication with N. This is the doubling function used in the Devaney reference, pp. 24-25, 27, 125. 132, 171,289.

Each cycle in row n can also be used to find from the base 2 version of its first entry (the seed s = s(N, i)) divided by N the other entries by repeated application of a cyclic left shift by one step (called sigma operation) to the period of the base 2 expression of s/N. E.g., n = 7, N = 15, P(N) = 4, s = 1: (1/15){10->2} = .repeat(0001), then (.repeat(0010)){2->10} = 2/10, (.repeat(0100)){2->10} = 4/10 and (.repeat(1000)){2->10} = 8/15. Similarly for s = 7: from (7/15)_{10->2} = .repeat(0111) one obtains by repeated sigma operations 14/15, 13/15 and 11/15. The proof uses the elementary formulas for the conversion from base 10 to base 2, and the reverse one, from base 2 to base 10. See also a comment on the period length P(N) given in A002326. (End)