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A323184 Numerators of rationals whose continued fraction representations show the prime factors of n (for n>1) in nondecreasing order. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 5 20:00 EDT 2022. Contains 357261 sequences. (Running on oeis4.)