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A334080 Numbers of Pythagorean triples contained in the divisors of the multiples of 60. 2
1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 3, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 8, 3, 6, 4, 6, 2, 8, 2, 6, 4, 5, 4, 9, 2, 4, 6, 8, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 9, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 7, 6, 8, 2, 8, 4, 9, 2, 12, 2, 4, 6, 6, 4, 12, 2, 10, 5, 4, 2, 12, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The odd numbers of the sequence are rare (see the table below).

The subsequence of odd terms begins with 1, 3, 3, 3, 5, 3, 5, 9, 3, 9, 7, 9, 5, 9, 9, 3, 11, 15, 5, 9, 5, 15, 9, 9, 9, 5, 19, 3, 15, 15, 9,...(see the table in the link).

It is interesting to note that each set of divisors of A169823(n) contains m primitive Pythagorean triples for some n, m = 1, 2,...

Examples:

- The set of divisors of A169823(1)= 60 contains only one primitive Pythagorean triple: (3, 4, 5).

- The set of divisors of A169823(136) = 8160 contains two primitive Pythagorean triples: (3, 4, 5) and (8, 15, 17).

- The set of divisors of A169823(910) = 54600 contains three primitive Pythagorean triples: (3, 4, 5), (5, 12, 13) and (7, 24, 25).

There is an interesting property: we observe that a(n) = A000005(n) except for n in the set {13, 26, 34, 39, 52, 65, 68, 70, 78, 91, 102, ...}. This set contains subset of numbers of the form 13*k, 34*k, 70*k, 203*k, 246*k, 259*k,... for k = 1, 2, ...

We recognize the sequence A081752: {13, 34, 70, 203, 246, 259, 671,...} (ordered product of the sides of primitive Pythagorean triangles divided by 60).

The following table shows the numbers of odd terms < 10^k for k = 2, 3, 4, 5, 6 and 7. For instance, among the 16 multiples of 60 less than 10^3, the divisors of the five numbers 60, 240, 540, 780 and 960 contain 1, 3, 3, 3 and 5 Pythagorean triples respectively, and that represents 31.25% of odd numbers.

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

|   Intervals   |    Number of    | Number of odd terms |           |

| D(k) < 10^k   | multiples of 60 |      in D(k)        |     %     |

| k = 2,3,...,7 |    in D(k)      |                     |           |

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

|   < 10^2      |          1      |          1          | 100 %     |

|   < 10^3      |         16      |          5          |  31.250 % |

|   < 10^4      |        166      |         18          |  10.843 % |

|   < 10^5      |       1666      |         72          |   4.321 % |

|   < 10^6      |      16666      |        256          |   1.536 % |

|   < 10^7      |     166666      |        879          |   0.527 % |

|---------------+-----------------+---------------------+-----------+

LINKS

Table of n, a(n) for n=1..85.

Michel Lagneau, Odd terms

EXAMPLE

a(4) = 3 because the divisors of A169823(4)= 240 are {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240} with 3 Pythagorean triples: (3, 4, 5), (6, 8, 10) and (12, 16, 20). The first triple is primitive.

MAPLE

with(numtheory):

for n from 60 by 60 to 5400 do :

   d:=divisors(n):n0:=nops(d):it:=0:

    for i from 1 to n0-1 do:

     for j from i+1 to n0-2 do :

      for m from i+2 to n0 do:

       if d[i]^2 + d[j]^2 = d[m]^2

        then

        it:=it+1:

        else

       fi:

      od:

     od:

    od:

    printf(`%d, `, it):

   od:

PROG

(PARI) ishypo(n) = setsearch(Set(factor(n)[, 1]%4), 1); \\ A009003

a(n) = {n *= 60; my(d=divisors(n), nb=0); for (i=3, #d, if (ishypo(d[i]), for (j=2, i-1, for (k=3, j-1, if (d[j]^2 + d[k]^2 == d[i]^2, nb++); ); ); ); ); nb; } \\ Michel Marcus, Apr 26 2020

CROSSREFS

Cf. A000005, A009003, A014442, A081752, A089120, A144255, A169823.

Sequence in context: A122668 A073668 A302051 * A066800 A218705 A193459

Adjacent sequences:  A334077 A334078 A334079 * A334081 A334082 A334083

KEYWORD

nonn

AUTHOR

Michel Lagneau, Apr 14 2020

STATUS

approved

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Last modified April 11 19:49 EDT 2021. Contains 342888 sequences. (Running on oeis4.)