[seqfan] Pairs Occurring Only Once Among # Of Divisors

Leroy Quet q1qq2qqq3qqqq at yahoo.com
Thu Jun 11 15:23:00 CEST 2009

[Sorry if this appears twice.]

I wonder if someone can run a program to determine possible values for my sequence.

For instance, the program would make sure a pair occurs only once among the number-of-divisors of all positive integers < some big number.

Then the values can be independently proved to be unique, which shouldn't be hard for most of them, hopefully.

Leroy Quet

I wrote:

I just submitted this sequence:

%I A161460
%S A161460 1,2,3,4,8,15,16,24
%N A161460 Those positive integers n such that there is no m different than n where both d(n) = d(m) and d(n+1) = d(m+1), where d(n) is the number of positive divisors of n. 
%e A161460 d(15) = 4, and d(15+1) = 5. Any positive integers m+1 with exactly 5 divisors must by of the form p^4, where p is prime. So m = p^4 -1 = (p^2+1)*(p+1)*(p-1). Now, in order for d(m) to have exactly 4 divisors, m must either be of the form q^3 or q*r, where q and r are distinct primes. But no p is such that (p^2+1)*(p+1)*(p-1) = q^3. And the only p where (p^2+1)*(p+1)*(p-1) = q*r is when p = 2 ( and so q=5, r =3). So, there is only one m where both d(m) = 4 and d(m+1) = 5, which is when m=15. Therefore, 15 is in this sequence. 
%K A161460 more,nonn
%O A161460 1,2


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