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 A289815 The first of a pair of coprime numbers whose factorizations depend on the ternary representation of n (see Comments for precise definition). 5
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 floor, 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=floor(n/3)         o+=1 print map(a, range(101)) # Indranil Ghosh, Aug 02 2017 CROSSREFS Cf. A000961, A004488, A289272, A289813, A289816, A289838, A289905. Sequence in context: A335444 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

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Last modified January 17 03:14 EST 2021. Contains 340214 sequences. (Running on oeis4.)