Sloane's A031122

Klaus Brockhaus klaus-brockhaus at t-online.de
Sat Oct 20 16:25:22 CEST 2001

```Don Reble schrieb:
>
> Numseq fans, Dan Hoey:
>
> > ID Number: A031122
> > Sequence:  17,48,72,122,305,323,329,842,1292,1353,2208,2255,5473,5777,
> >            5796,10946
> > Name:      Integers that appear as ratios of Fibonacci numbers
> >            F(kn)/F(k), but omitting Fibonacci numbers F(n)/F(1) and
> >            Lucas numbers F(2n)/F(n)
> > Author(s): Dan Hoey (Hoey at AIC.NRL.Navy.Mil)
>
>     I tried to extend this one, and found that 10946 doesn't belong.
>     Would a third person please check?
>
>     Here's my version.
>
>         17 48 72 122 305 323 329 842 1292 1353 2208 2255 5473 5777
>         5796 15005 15128 15456 23184 24447 39602 98209 103683 103729
>         104005 105937 166408 271442 416020 439128 709805 710648
>         726103 1762289 1845493 1860497 1860621 1866294 4870848
>         4873055 4976784 7465176 7880997 12752042 20466831 20608792
>         31622993 33379505 33385283 33385604 33489287 34111385
>         87403802 133957148 141421803 226980634 228826128 228841255
>         228929856 233802911 567451585 598364773 599074577 599075421
>         600940872
>
> --
> Don Reble       djr at nk.ca

Here is the result of a (rather crude) PARI program:

? a031122(mk,mn,x) =

local(v,vv,j,i);v=[];for(k=3,mk,for(n=3,mn,v=concat(v,[fibonacci(k*n)/fibonacci(k)])));v=vecsort(v);vv=[v[1]];for(j=2,matsize(v)[2],if(v[j-1]<v[j]&&v[j]<x,vv=concat(vv,[v[j]])));i=1;vv
time = 0 ms.
? a031122(150,50,610000000)
time = 2mn, 15,780 ms.
%19 = [17, 48, 72, 122, 305, 323, 329, 842, 1292, 1353, 2208, 2255,
5473, 5777, 5796, 15005, 15128, 15456, 23184, 24447, 39602, 98209,
103683, 103729, 104005, 105937, 166408, 271442, 416020, 439128, 709805,
710648, 726103, 1762289, 1845493, 1860497, 1860621, 1866294, 4870848,
4873055, 4976784, 7465176, 7880997, 12752042, 20466831, 20608792,
31622993, 33379505, 33385283, 33385604, 33489287, 34111385, 87403802,
133957148, 141421803, 226980634, 228826128, 228841255, 228929856,
233802911, 567451585, 598364773, 599074577, 599075421, 600940872]

It seems to be in complete agreement with your version, so 10946
obviously doesn't belong in the sequence.

Klaus Brockhaus

```