

A333948


Number of integersided triangles with perimeter n whose side lengths have the same number of divisors.


0



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
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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((nk)/2)} sign(floor((i+k)/(nik+1))) * [d(k) = d(i) = d(nik)], 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



