A120249 Numerator of cfenc[n] (see definition in comments).

1, 2, 3, 3, 5, 5, 8, 4, 4, 8, 13, 7, 21, 13, 7, 5, 34, 7, 55, 11, 11, 21, 89, 9, 7, 34, 5, 18, 144, 12, 233, 6, 18, 55, 12, 10, 377, 89, 29, 14, 610, 19, 987, 29, 9, 144, 1597, 11, 11, 11, 47, 47, 2584, 9, 19, 23, 76, 233, 4181, 17, 6765, 377, 14, 7, 31, 31, 10946, 76, 123, 19

Offset

1,2

Comments

a[n] := numerator of cfenc[n]. cfenc[n] := number given by interpreting as a continued fraction expansion (indexed from 1) the sequence whose i-th entry is one plus the exponent on the i-th prime factor of n (fix cfenc[1]=1). a[2^k] = cfenc[2^k] = k+1.

Links

Hans Havermann, Table of n, a(n) for n = 1..10000

Wikipedia, Continued fraction

Formula

a(2^k) = k+1.

a(A000040(n)) = A000045(n+2).

From Antti Karttunen, Oct 29 2019: (Start)

The following formula employs Gauss's notation for continued fractions (see the section "Notations" in the Wikipedia-article), for example, K_{i=1..3} u(i) stands for 1/(u(1) + 1/(u(2) + 1/u(3))):

Let c(n) = A001511(n) + K_{i=2..A061395(n)} 1/(1+A286561(n,A000040(i))). Then a(n) is the numerator of c(n), and A120250(n) is the denominator of c(n).

For all n >= 2, a(2n) = a(A003961(n)), thus a(n) = f(A323080(n)) for some function f.

(End)

Example

a(2646) = numerator[cfenc[2646]]= numerator[cfenc[2^1 * 3^3 * 7^2]] = numerator[FromContinuedFraction[{2; 4, 1, 3}]] = numerator[2 + 1/(4 + 1/(1 + 1/3))] = numerator[42/19] = 42.
From Antti Karttunen, Oct 29 2019: (Start)
a(6) = 3 because 6 = 2^1 * 3^1, and the numerator of the continued fraction 1+1 + 1/(1+1) = 5/2 is 5.
a(12) = 7 because 12 = 2^2 * 3^1, and the numerator of the continued fraction 2+1 + 1/(1+1) = 7/2 is 7.
a(15) = 7 because 15 = 2^0 * 3^1 * 5^1, and the numerator of the continued fraction 0+1 + 1/(1+1 + 1/(1+1)) = 1 + 1/(2 + 1/2) = 1 + 2/5 = 7/5 is 7.
(End)

Mathematica

Table[If[n == 1, 1, (fl = FactorInteger[n]; pq = Table[1, {i, 1, PrimePi[Last[fl][[1]]]}]; While[Length[fl] > 0, pp = First[fl]; fl = Drop[fl, 1]; pq[[PrimePi[pp[[1]]]]] = pp[[2]] + 1; ]; Numerator[FromContinuedFraction[pq]])], {n, 1, 80}]

Prog

(PARI) A120249(n) = if(1==n, n, my(pi=primepi(vecmax(factor(n)[, 1])), cf=1+valuation(n, prime(pi))); pi--; while(pi, cf = (1+valuation(n, prime(pi)))+(1/cf); pi--); numerator(cf)); \\ Antti Karttunen, Oct 26 2019

Crossrefs

Corresponding denominators in A120250. Numerators modulo respective denominators in A120251.

Cf. A000040, A000045, A001511, A003961, A061395, A286561, A323080.

Cf. also A007305, A075158, A273671, A273672.

Sequence in context: A130006 A099609 A350865 * A058690 A290369 A087153

Adjacent sequences: A120246 A120247 A120248 * A120250 A120251 A120252

Keyword

frac,nonn

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), Jun 12 2006, Jun 25 2006

Status

approved