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A339857
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Smallest side of integer-sided primitive triangles whose sides a < b < c form a geometric progression.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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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.
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MAPLE
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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;
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PROG
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(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
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CROSSREFS
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Cf. A336751 (similar for sides in arithmetic progression).
Cf. A335894 (similar for angles in arithmetic progression).
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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