A390223 Numbers k such that for the k-th almost-equilateral Heronian triangle (triangle with consecutive integer sides and integer area), perimeter - inradius is a prime or semiprime.

1, 2, 3, 4, 5, 7, 10, 13, 19, 25, 35, 38, 43, 50, 55, 79, 129, 133, 134, 169, 205, 208, 374

Offset

1,2

Comments

Numbers k >= 1 such that A001222(3 * A003500(k) - A001353(k)) <= 2.

Links

Table of n, a(n) for n=1..23.

Formula

The perimeter - inradius of the k-th triangle is b(k) = 3 * A003500(k) - A001353(k), which satisfies the recurrence b(k) = 4*b(k-1) - b(k-2).

Example

a(1) = 1 is a term because 3 * A003500(1) - A001353(1) = 3 * 4 - 1 = 11 is prime, corresponding to the triangle of sides 3,4,5 with inradius 1.
a(6) = 7 is a term because 3 * A003500(7) - A001353(7) = 3 * 10084 - 2911 = 27341 = 19 * 1439 is a semiprime, corresponding to the triangle of sides 10083, 10084, 10085 with inradius 2911.

Maple

q:= proc(n) option remember; 4*procname(n-1)-procname(n-2) end proc:
q(1):= 11: q(2):= 38:
select(i -> NumberTheory:-Omega(q(i))<=2, [$1..150]);

Crossrefs

Cf. A001222, A001353, A003500.

Sequence in context: A271489 A018127 A017835 * A007601 A195944 A087830

Adjacent sequences: A390220 A390221 A390222 * A390224 A390225 A390226

Keyword

nonn,more

Author

Will Gosnell and Robert Israel, Jan 19 2026

Extensions

a(20)-a(23) from Sean A. Irvine, Jan 24 2026

Status

approved