[seqfan] a(n) divides A000201(a(n))^m

Paul D Hanna pauldhanna at juno.com
Fri Jan 28 23:13:27 CET 2011


SeqFans, 
      Would someone extend/verify these sequences?  
Below, a(n) refers to the terms in SEQ.1. 
 
SEQ. (1)  
a(n) divides A000201(a(n))^m for some integer m>0, where A000201 is the lower Wythoff sequence. 
 
1,4,8,25,50,108,169,243,256,338,486,512,729,768,972,1024,1156,1215,
2312,3375,5000,7921,8192,8748,10000,12800,15000,15842,20000,25000,
 
Many of these terms are powers of fibonacci numbers. 
Perhaps this is expected, since these involve floor(a(n)*phi). 
  
SEQ. (2) 
A000201(a(n))
 
1,6,12,40,80,174,273,393,414,546,786,828,1179,1242,1572,1656,1870,1965,
3740,5460,8090,12816,13254,14154,16180,20710,24270,25632,32360,40450,
  
SEQ. (3) 
Least m for which a(n) divides A000201(a(n))^m. 
 
1,2,2,2,2,3,2,5,8,2,5,5,3,8,5,4,2,5,2,3,4,2,13,7,4,9,4,2,4,3,
 
Question: is m always <= n ? 
 
Thanks, 
     Paul 



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