OFFSET
1,7
COMMENTS
Terms only depends on prime signature of n^2+1. - David A. Corneth, Nov 14 2018
We observe an interesting statistic for n <= 10^5: the four values of a(n) = 0, 1, 6, 25 represent more than 82% (see the table below).
In the general case, a(k) = m if k^2+1 = p*q^m, m = 1, 2, 3, ... with p, q primes.
+--------------+-----------------------+------------+
| | number of occurrences | |
| a(n) | for n <= 10^5 | percentage |
+--------------+-----------------------+------------+
| 0 | 6656 | 6.656% |
| 1 | 23255 | 23.255% |
| 6 | 31947 | 31.947% |
| 25 | 20461 | 20.461% |
| other values | 17681 | 17.681% |
+--------------+-----------------------+------------+
FORMULA
a(n) = A089233(n^2+1). - Michel Marcus, Nov 13 2018
EXAMPLE
a(13) = 6 because the divisors {d(i)} of 13^2 + 1 = 170 (without the number 1) are {2, 5, 10, 17, 34, 85, 170}, and gcd(d(i), d(j)) = 1 for the 6 following pairs of elements of {d(i)}: (2, 5), (2, 17), (2, 85), (5, 17), (5, 34) and (10, 17).
MAPLE
with(numtheory):nn:=10^3:
for n from 1 to nn do:
it:=0:d:=divisors(n^2+1):n0:=nops(d):
for k from 2 to n0-1 do:
for l from k+1 to n0 do:
if gcd(d[k], d[l])= 1
then
it:=it+1
else
fi:
od:
od:
printf(`%d, `, it):
od:
MATHEMATICA
f[n_] := (DivisorSigma[0, n^2] - 1)/2 - DivisorSigma[0, n] + 1; Map[f, Range[0, 100]^2+1] (* Amiram Eldar, Nov 14 2018 after Robert G. Wilson v at A089233 *)
PROG
(PARI) a(n) = {my(d=divisors(n^2+1)); sum(k=2, #d, sum(j=2, k-1, gcd(d[k], d[j]) == 1)); } \\ Michel Marcus, Nov 12 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Michel Lagneau, Nov 12 2018
STATUS
approved