A127611 a(n) = numerator of the continued fraction which has the positive divisors of n as its terms.

1, 3, 4, 13, 6, 63, 8, 107, 37, 163, 12, 3259, 14, 311, 319, 1725, 18, 10449, 20, 13928, 613, 751, 24, 638475, 151, 1043, 1003, 37306, 30, 1513023, 32, 55307, 1489, 1771, 1511, 19381852, 38, 2207, 2071, 4538318, 42, 5649833, 44, 142046, 131413, 3223, 48

Offset

1,2

Comments

The divisors can be written either from largest to smallest or from smallest to largest and the numerator of the continued fraction would remain unchanged.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Formula

If p is prime, a(p^k) = p^k * a(p^(k-1)) + a(p^(k-2)), with a(p^0) = a(1) = 1 and a(p^1) = p+1. - Robert Israel, Jan 17 2023

Example

The divisors of 6 are 1,2,3,6. So a(6) is the numerator of 1 +1/(2 +1/(3 +1/6)) = 63/44. a(6) is also the numerator of 6 +1/(3 +1/(2+1/1)) = 63/10.

Maple

f:= n -> numer(numtheory:-cfrac(sort(convert(numtheory:-divisors(n), list)))):
map(f, [$1..100]); # Robert Israel, Jan 17 2023

Mathematica

f[n_] := Numerator[FromContinuedFraction[Divisors[n]]]; Table[f[n], {n, 47}] (* Ray Chandler, Jan 22 2007 *)

Prog

(PARI) a(n) = contfracpnqn(divisors(n))[1, 1]; \\ Kevin Ryde, Jan 19 2023

Crossrefs

Cf. A127612 (denominators), A127613 (denominators).

Cf. A061377, A280219.

Sequence in context: A327583 A324159 A220847 * A308688 A324501 A359112

Adjacent sequences: A127608 A127609 A127610 * A127612 A127613 A127614

Keyword

frac,nonn,look

Author

Leroy Quet, Jan 19 2007

Extensions

Extended by Ray Chandler, Jan 22 2007

Status

approved