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A321519 Let d(n,i), i = 1..k be the k divisors of n^2 + 1 (the number 1 is not counted). a(n) is the number of ordered pairs d(n,i) < d(n,j) such that gcd(d(n,i), d(n,j)) = 1. 0
0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 1, 1, 6, 0, 1, 0, 6, 2, 1, 0, 6, 1, 6, 0, 1, 0, 6, 1, 1, 1, 6, 2, 6, 1, 1, 0, 6, 2, 1, 0, 2, 1, 11, 1, 1, 1, 25, 1, 1, 1, 1, 1, 6, 0, 6, 0, 16, 1, 1, 1, 1, 1, 6, 1, 1, 0, 6, 3, 1, 2, 1, 6, 25, 0, 6, 1, 6, 1, 1, 1, 6, 2, 25, 0, 1, 1 (list; graph; refs; listen; history; text; internal format)
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).

a(A005574(n)) = 0, a(A085722(n)) = 1, a(A272078(n)) = 6, a(A316351(n)) = 25.

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%  |

+--------------+-----------------------+------------+

LINKS

Table of n, a(n) for n=1..86.

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

Cf. A005574, A002522, A085722, A089233, A193432, A272078, A316351.

Sequence in context: A318163 A114640 A056890 * A346392 A169590 A300185

Adjacent sequences:  A321516 A321517 A321518 * A321520 A321521 A321522

KEYWORD

nonn

AUTHOR

Michel Lagneau, Nov 12 2018

STATUS

approved

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Last modified January 24 08:48 EST 2022. Contains 350534 sequences. (Running on oeis4.)