A020890 -id:A020890 - cp's OEIS frontend

A343894 Perimeters of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c. The triples (a, b, c) are listed in increasing order of side a, and if sides a coincide, in increasing order of side b.

13, 37, 47, 71, 73, 107, 121, 143, 183, 177, 181, 191, 241, 239, 249, 253, 291, 299, 347, 337, 359, 409, 421, 429, 431, 433, 491, 517, 503, 529, 563, 537, 541, 579, 587, 649, 659, 661, 671, 753, 743, 769, 759, 781, 831, 767, 789, 793, 897, 851, 923, 863, 913, 947, 1033, 933

Offset: 1

Bernard Schott, May 07 2021

nonn

The triples (a, b, c) are listed in increasing order of side a, and if sides a coincide then in increasing order of side b.
The sequence is not monotonic: a(9) = 183 > a(10) = 177.
All terms are odd.
For the corresponding primitive triples and miscellaneous properties and references, see A343891.

			a(3) = 15 + 12 + 20 = 47, because the third triple is (15, 12, 20) with relations 2/15 = 1/12 + 1/20 and 20-15 < 12 < 20+15.

Michel Marcus, Table of n, a(n) for n = 1..1000

Cf. A343891 (triples), A020883 (side a), A343892 (side b), A343893 (side c), A343895.

Cf. A020886, A020890.

for a from 4 to 200 do
for b from floor(a/2)+1 to a-1 do
c := a*b/(2*b-a);
if c=floor(c) and igcd(a,b,c)=1 and c-b

PARI

lista(nn) = {my(list = List()); for (a=4, nn, for (b = floor(a/2)+1, a-1, my(c = a*b/(2*b-a)); if ((denominator(c) == 1) && (gcd([a, b, c]) == 1) && (c-bMichel Marcus, May 10 2021

a(n) = A343891(n, 1) + A343891(n, 2) + A343891(n, 3).

a(n) = A020883(n) + A343892(n) + A343893(n).

A343892 Side b of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.

Original entry on oeis.org

3, 10, 12, 15, 21, 30, 36, 35, 40, 44, 55, 56, 52, 63, 65, 78, 70, 90, 77, 105, 99, 85, 102, 119, 132, 136, 117, 114, 143, 133, 126, 152, 171, 154, 182, 168, 165, 210, 195, 161, 176, 184, 208, 207, 187, 240, 230, 253, 200, 221, 198, 255, 225, 234, 216, 275, 300, 306, 247, 270

Offset: 1

Bernard Schott, May 06 2021

nonn

The triples (a, b, c) are displayed in increasing order of side a, and if sides a coincide then in increasing order of the side b.
The sequence is not increasing because a(7) = 36 > a(8) = 35, but, these sides b are listed in increasing order in A020890.
The first term appearing twice is 330 and corresponds to triples (435, 330, 638) and (460, 330, 759), the second one is 462 and corresponds to triples (483, 462, 506) and (532, 462, 627).
For the corresponding primitive triples and miscellaneous properties and references, see A343891.

			a(4) = 15, because the fourth triple is (21, 15, 35) with side b = 15, satisfying 1/15 = 2/21 - 1/35 and 31-15 < 21 < 31+15.

Cf. A343891 (triples), A020883 (side a), A343893 (side c), A343894 (perimeter).

Cf. A020890 (sides b ordered).

for a from 4 to 200 do
for b from floor(a/2)+1 to a-1 do
c := a*b/(2*b-a);
if c=floor(c) and igcd(a,b,c)=1 and c-b

a(n) = A343891(n, 2).

A020889 Ordered set of b + c - a as (a,b,c) runs through all primitive Pythagorean triples with a

Original entry on oeis.org

6, 20, 24, 30, 42, 60, 70, 72, 80, 88, 104, 110, 112, 126, 130, 140, 154, 156, 170, 180, 198, 204, 210, 228, 234, 238, 252, 264, 266, 272, 286, 304, 308, 322, 330, 336, 342, 352, 364, 368, 374, 390, 396, 400, 414, 416, 420, 432, 442, 450, 460, 468, 480, 494, 506, 510

Offset: 1

Clark Kimberling

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A020890.

a(n) = 2*A020890(n).

Extended and corrected by David W. Wilson

Offset corrected to 1 by Ray Chandler, Jan 23 2020

A343895 Perimeters of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the two other sides b and c, i.e., 2/a = 1/b + 1/c with b < a.

Original entry on oeis.org

13, 37, 47, 71, 73, 107, 121, 143, 177, 181, 183, 191, 239, 241, 249, 253, 291, 299, 337, 347, 359, 409, 421, 429, 431, 433, 491, 517, 503, 529, 537, 541, 563, 579, 587, 649, 659, 661, 671, 743, 753, 759, 767, 769, 781, 789, 793, 831, 851, 897, 863, 913, 923, 933, 937, 947, 971, 1033

Offset: 1

Bernard Schott, May 18 2021

nonn

This sequence is the list of ordered terms of A343894, which is not monotonic.
It first differs from A343894 at index 9 where a(9) = 177 while A343894(9) = 183.
Like A343894, all terms are odd.
For the corresponding primitive triples and miscellaneous properties and references, see A343891.

			a(2) = 12 + 10 + 15 = 37, because the second largest perimeter corresponds to triple (12, 10, 15) with relations 2/12 = 1/10 + 1/15 and 15 - 10 < 12 < 15 + 10.

Cf. A343894.
Cf. A343891, A343892, A343893.
Cf. A020883 (sides a), A020886 (sides c), A020890 (sides b).

cp's OEIS Frontend

A343894 Perimeters of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c. The triples (a, b, c) are listed in increasing order of side a, and if sides a coincide, in increasing order of side b.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

PARI

Formula

A343892 Side b of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Maple

Formula

A020889 Ordered set of b + c - a as (a,b,c) runs through all primitive Pythagorean triples with a

Views

Author

Keywords

Links

Crossrefs

Formula

Extensions

A343895 Perimeters of integer-sided primitive triangles (a, b, c) where side a is the harmonic mean of the two other sides b and c, i.e., 2/a = 1/b + 1/c with b < a.

Views

Author

Keywords

Comments

Examples

Crossrefs