A289815 The first of a pair of coprime numbers whose factorizations depend on the ternary representation of n (see Comments for precise definition).

1, 2, 1, 3, 6, 3, 1, 2, 1, 4, 10, 5, 12, 30, 15, 4, 10, 5, 1, 2, 1, 3, 6, 3, 1, 2, 1, 5, 14, 7, 15, 42, 21, 5, 14, 7, 20, 70, 35, 60, 210, 105, 20, 70, 35, 5, 14, 7, 15, 42, 21, 5, 14, 7, 1, 2, 1, 3, 6, 3, 1, 2, 1, 4, 10, 5, 12, 30, 15, 4, 10, 5, 1, 2, 1, 3, 6

Offset

0,2

Comments

For n >= 0, with ternary representation Sum_{i=1..k} t_i * 3^e_i (all t_i in {1, 2} and all e_i distinct and in increasing order):

- let S(0) = A000961 \ { 1 },

- and S(i) = S(i-1) \ { p^(f + j), with p^f = the (e_i+1)-th term of S(i-1) and j > 0 } for any i=1..k,

- then a(n) = Product_{i=1..k such that t_i=1} "the (e_i+1)-th term of S(k)".

See A289816 for the second coprime number.

See A289838 for the product of this sequence with A289816.

By design, gcd(a(n), A289816(n)) = 1.

Also, the number of distinct prime factors of a(n) equals the number of ones in the ternary representation of n.

We also have a(n) = A289816(A004488(n)) for any n >= 0.

For each pair of coprime numbers, say x and y, there is a unique index, say n, such that a(n) = x and A289816(n) = y; in fact, n = A289905(x,y).

This sequence combines features of A289813 and A289272.

The scatterplot of the first terms of this sequence vs A289816 (both with logarithmic scaling) looks like a triangular cristal.

For any t > 0: we can adapt the algorithm used here and in A289816 in order to uniquely enumerate every tuple of t mutually coprime numbers (see Links section for corresponding program).

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10000

Rémy Sigrist, Scatterplot of the first 10000 terms of A289815 vs A289816 (both with logarithmic scaling)

Rémy Sigrist, PARI program to uniquely enumerate tuples of mutually coprime numbers

Formula

a(A005836(n)) = A289272(n-1) for any n > 0.

a(2 * A005836(n)) = 1 for any n > 0.

Example

For n=42:
- 42 = 2*3^1 + 1*3^2 + 1*3^3,
- S(0) = { 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, ... },
- S(1) = S(0) \ { 3^(1+j) with j > 0 }
       = { 2, 3, 4, 5, 7, 8,    11, 13, 16, 17, 19, 23, 25,     29, ... },
- S(2) = S(1) \ { 2^(2+j) with j > 0 }
       = { 2, 3, 4, 5, 7,       11, 13,     17, 19, 23, 25,     29, ... },
- S(3) = S(2) \ { 5^(1+j) with j > 0 }
       = { 2, 3, 4, 5, 7,       11, 13,     17, 19, 23,         29, ... },
- a(42) = 4 * 5 = 20.

Prog

(PARI) a(n) =
{
    my (v=1, x=1);
    for (o=2, oo,
        if (n==0, return (v));
        if (gcd(x, o)==1 && omega(o)==1,
            if (n % 3,    x *= o);
            if (n % 3==1, v *= o);
            n \= 3;
        );
    );
}
(Python)
from sympy import gcd, primefactors
def omega(n): return 0 if n==1 else len(primefactors(n))
def a(n):
    v, x, o = 1, 1, 2
    while True:
        if n==0: return v
        if gcd(x, o)==1 and omega(o)==1:
            if n%3: x*=o
            if n%3==1:v*=o
            n //= 3
        o+=1
print([a(n) for n in range(101)]) # Indranil Ghosh, Aug 02 2017

Crossrefs

Cf. A000961, A004488, A289272, A289813, A289816, A289838, A289905.

Sequence in context: A358646 A006895 A202204 * A125205 A125206 A221918

Adjacent sequences: A289812 A289813 A289814 * A289816 A289817 A289818

Keyword

nonn,base,look

Author

Rémy Sigrist, Jul 12 2017

Status

approved