A333948 Number of integer-sided triangles with perimeter n whose side lengths have the same number of divisors.

0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 2, 1, 1, 1, 1, 3, 1, 3, 2, 2, 4, 2, 3, 3, 4, 1, 3, 2, 5, 4, 4, 2, 6, 2, 5, 2, 6, 3, 6, 2, 6, 4, 7, 5, 7, 4, 6, 5, 6, 6, 8, 7, 9, 5, 5, 5, 11, 5, 9, 3, 8, 8, 13, 8, 10, 6, 6, 9, 15, 13, 13, 10, 15, 12, 16, 14, 17, 10

Offset

1,12

Links

Table of n, a(n) for n=1..84.

Wikipedia, Integer Triangle

Formula

a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} sign(floor((i+k)/(n-i-k+1))) * [d(k) = d(i) = d(n-i-k)], where [] is the Iverson bracket and d(n) is the number of divisors of n (A000005).

Example

a(12) = 2; The perimeter of each of the two triangles [2,5,5] and [4,4,4] is 12 and the side lengths for each triangle share the same number of divisors (i.e., d(2) = d(5) = d(5) = 2 and d(4) = d(4) = d(4) = 3).
a(15) = 2; The perimeter of each of the two triangles [3,5,7] and [5,5,5] is 15 and the side lengths for each triangle share the same number of divisors (i.e., d(3) = d(5) = d(7) = 2 and d(5) = d(5) = d(5) = 2).

Mathematica

Table[Sum[Sum[KroneckerDelta[DivisorSigma[0, i], DivisorSigma[0, k], DivisorSigma[0, n - i - k]]*Sign[Floor[(i + k)/(n - i - k + 1)]], {i, k, Floor[(n - k)/2]}], {k, Floor[n/3]}], {n, 100}]

Crossrefs

Cf. A000005, A005044.

Sequence in context: A236459 A190427 A287108 * A287360 A035443 A180430

Adjacent sequences: A333945 A333946 A333947 * A333949 A333950 A333951

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 14 2020

Status

approved