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

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, 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, 19, 29, 1, 37, 1, 31, 22, 70, 13, 34, 1, 35, 23, 36, 1, 56, 1, 37, 26

Offset

2,3

Comments

Denominators are given in A323185.

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)).

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).

The first few generating rationals are:

N ... GR .... continued fraction

2 ... 1/2 ... [0; 2]

3 ... 1/3 ... [0; 3]

4 ... 2/5 ... [0; 2, 2]

5 ... 1/5 ... [0; 5]

6 ... 3/7 ... [0; 2, 3]

7 ... 1/7 ... [0; 7]

8 ... 5/12 .. [0; 2, 2, 2]

9 ... 3/10 .. [0; 3, 3]

10 .. 5/11 .. [0; 2, 5]

a(n) is the numerator of the generating rational of n.

Iff n is prime, a(n) is 1.

Links

Georg Fischer, Table of n, a(n) for n = 2..1000

Example

a(28) = 15 because 15/37 = [0; 2, 2, 7] and 2*2*7 = 28.
a(29) = 1 because 1/29 = [0; 29] = 29.

Mathematica

Array[Numerator@ FromContinuedFraction@ Prepend[Flatten@ Map[ConstantArray[#1, #2] & @@ # &, FactorInteger@ #], 0] &, 74, 2] (* Michael De Vlieger, Jan 07 2019 *)

Prog

(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]))); }
A323184(n)={vectorise_factors(n); contfracpnqn(v)[1, 1]; }
for(k=2, 75, print1(A323184(k)", "))

Crossrefs

Cf. A323185.

Sequence in context: A214340 A283463 A283464 * A102614 A307149 A276421

Adjacent sequences: A323181 A323182 A323183 * A323185 A323186 A323187

Keyword

nonn,frac

Author

Chris Boyd, Jan 06 2019

Status

approved