login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A299068 Number of pairs of factors of n^2*(n^2-1) which differ by n. 2

%I #25 Feb 05 2018 02:55:21

%S 3,4,8,7,11,6,10,12,11,9,9,9,13,22,12,7,7,11,21,28,9,7,17,14,13,14,13,

%T 13,11,9,10,12,17,33,28,8,7,20,19,15,9,10,21,29,10,7,14,19,18,21,11,9,

%U 16,44,46,14,7,9,15,9,9,18,40,24,18,8,9,30,18,17,11

%N Number of pairs of factors of n^2*(n^2-1) which differ by n.

%C The question arose when seeking triples of numbers for which the sum of the squares of any two is congruent to 1 modulo the third.

%C From _Robert Israel_, Feb 04 2018: (Start)

%C For n > 7, a(n)>= 7, as there are at least the following pairs:

%C (1,n+1), (n,2*n), (2*n,3*n), ((n^2-n)/2,(n^2+n)/2), (n^2-n,n^2), (n^2,n^2+n), and (3*n, 4*n) (if n is odd) or (n/2,3*n/2) (if n is even).

%C If k in A299159 is sufficiently large, then a(12*k-2)=7. Dickson's conjecture implies there are infinitely many such k, and thus infinitely many n with a(n)=7. (End)

%H Robert Israel, <a href="/A299068/b299068.txt">Table of n, a(n) for n = 2..10000</a>

%p a:= n-> (s-> add(`if`(i+n in s, 1, 0), i=s))(

%p numtheory[divisors](n^2*(n^2-1))):

%p seq(a(n), n=2..100); # _Alois P. Heinz_, Feb 01 2018

%t Array[With[{d = Divisors[# (# - 1)] &[#^2]}, Count[d + #, _?(MemberQ[d, #] &)]] &, 71, 2] (* _Michael De Vlieger_, Feb 01 2018 *)

%Y Cf. A299159.

%K nonn

%O 2,1

%A _John H Mason_, Feb 01 2018

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 10 01:29 EDT 2024. Contains 371563 sequences. (Running on oeis4.)