A289816 The second of a pair of coprime numbers whose factorizations depend on the ternary representation of n (See Comments for precise definition).

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

Offset

0,3

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=2} "the (e_i+1)-th term of S(k)".

See A289815 for the first coprime number and additional comments.

The number of distinct prime factors of a(n) equals the number of twos in the ternary representation of n.

Links

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

Formula

a(n) = A289815(A004488(n)) for any n >= 0.

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

a(2 * A005836(n)) = A289272(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) = 3.

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==2, 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==2:v*=o
            n //= 3
        o+=1
print([a(n) for n in range(101)]) # Indranil Ghosh, Aug 02 2017

Crossrefs

Cf. A000961, A004488, A289815.

Sequence in context: A376939 A181572 A287731 * A374412 A363148 A054482

Adjacent sequences: A289813 A289814 A289815 * A289817 A289818 A289819

Keyword

nonn,base,look

Author

Rémy Sigrist, Jul 12 2017

Status

approved