cp's OEIS Frontend

A323184 Numerators of rationals whose continued fraction representations show the prime factors of n (for n>1) in nondecreasing order.

%I A323184 #21 Feb 20 2019 08:10:16
%S A323184 1,1,2,1,3,1,5,3,5,1,7,1,7,5,12,1,10,1,11,7,11,1,17,5,13,10,15,1,16,1,
%T A323184 29,11,17,7,23,1,19,13,27,1,22,1,23,16,23,1,41,7,26,17,27,1,33,11,37,
%U A323184 19,29,1,37,1,31,22,70,13,34,1,35,23,36,1,56,1,37,26
%N A323184 Numerators of rationals whose continued fraction representations show the prime factors of n (for n>1) in nondecreasing order.
%C A323184 Denominators are given in A323185.
%C A323184 From its prime factorization, each natural number N>1 can be uniquely represented as a tuple of nondecreasing first powers (e.g., 60 = 2*2*3*5 -> (2, 2, 3, 5)).
%C A323184 There is a unique positive finite continued fraction associated with N whose coefficients in the standard abbreviated notation (except for the first coefficient, which is arbitrarily set to zero) map 1-to-1 to the elements of the tuple, from which the corresponding generating rational can be calculated (e.g. 60 -> [0; 2, 2, 3, 5] = 37/90).
%C A323184 The first few generating rationals are:
%C A323184 N ... GR .... continued fraction
%C A323184 2 ... 1/2 ... [0; 2]
%C A323184 3 ... 1/3 ... [0; 3]
%C A323184 4 ... 2/5 ... [0; 2, 2]
%C A323184 5 ... 1/5 ... [0; 5]
%C A323184 6 ... 3/7 ... [0; 2, 3]
%C A323184 7 ... 1/7 ... [0; 7]
%C A323184 8 ... 5/12 .. [0; 2, 2, 2]
%C A323184 9 ... 3/10 .. [0; 3, 3]
%C A323184 10 .. 5/11 .. [0; 2, 5]
%C A323184 a(n) is the numerator of the generating rational of n.
%C A323184 Iff n is prime, a(n) is 1.
%H A323184 Georg Fischer, <a href="/A323184/b323184.txt">Table of n, a(n) for n = 2..1000</a>
%e A323184 a(28) = 15 because 15/37 = [0; 2, 2, 7] and 2*2*7 = 28.
%e A323184 a(29) = 1 because 1/29 = [0; 29] = 29.
%t A323184 Array[Numerator@ FromContinuedFraction@ Prepend[Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ #], 0] &, 74, 2] (* _Michael De Vlieger_, Jan 07 2019 *)
%o A323184 (PARI) vectorise_factors(m)={v=[0];F=factor(m);for(i=1,matsize(F)[1],for(j=1,F[i,2],v=concat(v,F[i,1])));}
%o A323184 A323184(n)={vectorise_factors(n); contfracpnqn(v)[1,1];}
%o A323184 for(k=2,75,print1(A323184(k)", "))
%Y A323184 Cf. A323185.
%K A323184 nonn,frac
%O A323184 2,3
%A A323184 _Chris Boyd_, Jan 06 2019