Permutations? Involving Division

Leroy Quet qq-quet at
Sun Apr 30 19:00:18 CEST 2006

I just submitted these two sequences.

>%S A118809 1,2,3,5,6,9,4,10,11,13,8,16,12,14
>%N A118809 a(1) = 1. a(n) = (number of earlier terms which divide n)th 
>positive integer not occurring among the earlier terms of the sequence.
>%C A118809 Likely a permutation of the positive integers.
>%e A118809 The first 7 terms of the sequence are 1,2,3,5,6,9,4. Of these 
>there are 3 terms (1, 2 and 4) that divide 8, so we want for a(8) the 3rd 
>positive integer not among the first 7 terms of the sequence (ie we want 
>the third term of 7,8,10,11,...). So a(8)= 10.
>%Y A118809 A118810
>%O A118809 1
>%K A118809 ,more,nonn,

>%S A118810 1,2,3,7,4,5,19,11,6,8,9,13,10,14
>%N A118810 Inverse permutation of sequence A118809.
>%C A118810 Is it certain that sequence A118809 is a permutation of the 
>positive integers?
>%Y A118810 A118809
>%O A118810 1
>%K A118810 ,more,nonn,

Can it be PROVED that these sequences are really permutations of (the 
sequence of every one of) the positive integers?

ie. Can we be sure that there is always an integer n, n > any arbitrary 
positive integer N, such that the first term of the sequence (which is 1) 
is the only term among the first (n-1) terms that divides n?

n can be a prime, for example. All we have to do to prove that the 
sequences are permutations of the positive integers then is prove that 
there are an infinite number of primes p that don't occur among the first 
(p-1) terms of the sequence.
(If, however, the unlikely situation occurs where, after some prime P, 
every prime p, p > P, occurs somewhere among the first (p-1) terms of the 
sequence, then the sequences will not likely be permutations of the 
positive integers.)

Leroy Quet

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