login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A339857 Smallest side of integer-sided primitive triangles whose sides a < b < c form a geometric progression. 4
4, 9, 16, 25, 25, 25, 36, 49, 49, 49, 49, 64, 64, 81, 81, 81, 81, 100, 100, 121, 121, 121, 121, 121, 121, 144, 144, 144, 169, 169, 169, 169, 169, 169, 169, 169, 196, 196, 196, 225, 225, 225, 225, 225, 256, 256, 256, 256, 256, 289, 289, 289, 289, 289, 289, 289, 289, 289, 289 (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 a's.
All the terms are the squares >= 4 in increasing order.
For the corresponding primitive triples and miscellaneous properties, see A339856.
LINKS
FORMULA
a(n) = A339856(n, 1).
EXAMPLE
a(1) = 4 for only the smallest such triangle (4, 6, 9).
a(4) = 25 for triple (25, 30, 36) with 25 * 36 = 30^2 and ratio q_1 = 6/5, hence for this triangle, C < Pi/2 because 1 < q_1 = 6/5 < sqrt(phi)); also a(5) = 25 for the triple (25, 35, 49) with 25 * 49 = 35^2 and ratio q_2 = 7/5; then a(6) = 25 for the triple (25, 40, 64) with 25*64 = 40^2 and ratio q_3 = 8/5, hence, for these two last triangles, C > Pi/2 because sqrt(phi) < q_2 < q_3 < phi.
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(a); 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(a, ", ")); ); ); ); } \\ Michel Marcus, Dec 26 2020
CROSSREFS
Cf. A339856 (triples), this sequence (smallest side), A339858 (middle side), A339859 (largest side), A339860 (perimeter).
Cf. A336751 (similar for sides in arithmetic progression).
Cf. A335894 (similar for angles in arithmetic progression).
Sequence in context: A259602 A017668 A225004 * A074373 A067115 A061077
KEYWORD
nonn
AUTHOR
Bernard Schott, Dec 25 2020
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 15:28 EDT 2024. Contains 371254 sequences. (Running on oeis4.)