A299159 -id:A299159 - cp's OEIS frontend

A299068 Number of pairs of factors of n^2*(n^2-1) which differ by n.

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

Offset: 2

John H Mason, Feb 01 2018

nonn

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.
From Robert Israel, Feb 04 2018: (Start)
For n > 7, a(n)>= 7, as there are at least the following pairs:
(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).
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)

Robert Israel, Table of n, a(n) for n = 2..10000

Cf. A299159.

a:= n-> (s-> add(`if`(i+n in s, 1, 0), i=s))(
        numtheory[divisors](n^2*(n^2-1))):
seq(a(n), n=2..100);  # Alois P. Heinz, Feb 01 2018

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

cp's OEIS Frontend

A299068 Number of pairs of factors of n^2*(n^2-1) which differ by n.

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