|
|
A339858
|
|
Middle side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.
|
|
4
|
|
|
6, 12, 20, 30, 35, 40, 42, 56, 63, 70, 77, 72, 88, 90, 99, 117, 126, 110, 130, 132, 143, 154, 165, 176, 187, 156, 204, 228, 182, 195, 208, 221, 234, 247, 260, 273, 210, 238, 266, 240, 255, 285, 330, 345, 272, 304, 336, 368, 400, 306, 323, 340, 357, 374, 391, 408, 425, 442, 459
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
The triples of sides (a, b, c) with a < b < c are in increasing lexicographic order. This sequence lists the b's.
For the corresponding primitive triples and miscellaneous properties and references, see A339856.
This sequence is not increasing. For example, a(11) = 77 for triple (49, 77, 121) while a(12) = 72 for triple (64, 72, 81).
Oblong numbers k*(k+1) >= 6 form a subsequence (A002378) and belong to triples of the form (k^2, k*(k+1), (k+1)^2).
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
a(1) = 6 only for the smallest such triangle (4, 6, 9) with 6^2 = 4*9 and a ratio q = 3/2.
a(2) = 12 only for the triangle (9, 12, 16) with 12^2 = 9*16 and a ratio q = 4/3.
|
|
MAPLE
|
for a from 1 to 300 do
for b from a+1 to floor((1+sqrt(5))/2 *a) do
for c from b+1 to floor((1+sqrt(5))/2 *b) do k:=a*c;
if k=b^2 and igcd(a, b, c)=1 then print(b); end if;
end do;
end do;
end do;
|
|
PROG
|
(PARI) lista(nn) = {my(phi = (1+sqrt(5))/2); for (a=1, nn, for (b=a+1, floor(a*phi), for (c=b+1, floor(b*phi), if ((a*c == b^2) && (gcd([a, b, c])==1), print1(b, ", "); ); ); ); ); } \\ Michel Marcus, Dec 30 2020
|
|
CROSSREFS
|
Cf. A336751 (similar for sides in arithmetic progression).
Cf. A335894 (similar for angles in arithmetic progression).
Cf. A002378 \ {0,2} (a subsequence).
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|