A357277 -id:A357277 - cp's OEIS frontend

A121940 Product of the first n primes of the form 6k+1.

7, 91, 1729, 53599, 1983163, 85276009, 5201836549, 348523048783, 25442182561159, 2009932422331561, 194963444966161417, 20081234831514625951, 2188854596635094228659, 277984533772656967039693, 38639850194399318418517327, 5834617379354297081196116377

Offset: 1

Jonathan Vos Post, Sep 03 2006

For n>1, a(n) is the least positive integer that can be primitively represented as m^2+mn+n^2 with 0<=m<=n and gcd(m,n)=1 in exactly 2^(n-1) ways. - Ray Chandler, Oct 01 2007
From Bernard Schott, Mar 30 2021: (Start)
Also, for n >= 1, a(n) is the smallest positive integer m such that m^2 can be primitively represented as k^2-k*q+q^2 with 1 <= k < q and gcd(k,q)= 1 in exactly 2^n ways. For example (a(1))^2 = 7^2 = 3^2 - 3*8 + 8^2 = 5^2 - 5*8 + 8^2.
It follows that a(n) is the smallest middle side b that appears exactly 2^n times consecutively in the data of A335895, for integer-sided triangles whose angles A < B < C are in arithmetic progression. (End)
Also, a(n) is the smallest largest side c that appears exactly 2^(n-1) times consecutively in the data of A357277 for integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees. - Bernard Schott, Oct 21 2022

			a(4) = 53599 = 7 * 13 * 19 * 31.

Ray Chandler, Table of n, a(n) for n=1..100

Cf. A002476, A002110, A006278, A078586, A335895.

Greedy inverse of A005088.

Rest@FoldList[Times, 1, Select[6 Range[100] + 1, PrimeQ]] (* Ray Chandler, Oct 01 2007 *)

lista(nn) = {my(pr=1, list=List()); forprime(p=1, nn, if ((p%3) == 1, listput(list,pr *= p));); Vec(list);} \\ Michel Marcus, Jul 17 2020

a(n) = Product_{i=1..n} A002476(i).

Extended by Ray Chandler, Oct 01 2007

A357274 List of primitive triples for integer-sided triangles with angles A < B < C and C = 2*Pi/3 = 120 degrees.

Original entry on oeis.org

3, 5, 7, 7, 8, 13, 5, 16, 19, 11, 24, 31, 7, 33, 37, 13, 35, 43, 16, 39, 49, 9, 56, 61, 32, 45, 67, 17, 63, 73, 40, 51, 79, 11, 85, 91, 19, 80, 91, 55, 57, 97, 40, 77, 103, 24, 95, 109, 13, 120, 127, 23, 120, 133, 65, 88, 133, 69, 91, 139, 56, 115, 151, 25, 143, 157, 75, 112, 163, 15, 161, 169, 104, 105, 181

Offset: 1

Bernard Schott, Sep 22 2022

The only triangles with integer sides that have an angle equal to a whole number of degrees are triangles which have an angle of 60° (A335893), or an angle of 90° (A263728) or an angle of 120° as here (see Keith Selkirk link, p. 251).
The triples are displayed in nondecreasing order of largest side c, and if largest sides coincide then by increasing order of the smallest side a, hence, each triple (a, b, c) is in increasing order.
The corresponding metric relation between sides is c^2 = a^2 + a*b + b^2.
The triples (a, b, c) can be generated with integers u, v such that gcd(u,v) = 1 and 0 < v < u:
-> a = u^2 - v^2
-> b = 2*u*v + v^2
-> c = u^2 + u*v + v^2.
Note that side c cannot be even when the triple is primitive as here.
The (3, 5, 7) triangle is the only primitive triangle with a 120-degree angle and with its integer sides in arithmetic progression (A336750). This smallest triple is obtained for u = 2 and v = 1.
The Fermat point of these triangles is vertex C, then distance FA+FB+FC = CA+CB = b+a is an integer.
If (a,b,c) is a primitive 120-triple, then both (a,a+b,c) and (a+b,b,c) are 60-triples in A335893, see Emrys Read link, lemma 2 p. 302.

			Table of triples begins:
   3,  5,  7;
   7,  8, 13;
   5, 16, 19;
  11, 24, 31;
   7, 33, 37;
............
(7, 8, 13) is a triple for this sequence because from the law of cosines (see link), cos(C) = (7^2 + 8^2 - 13^2)/(2*7*8) = -1/2.

G. Julia, Triangles dont un angle mesure 120 degrés, Problème Capes (in French).
Emrys Read, On integer-sided triangles containing angles of 120° or 60°, Mathematical Gazette 90, July 2006, 299-305.
Keith Selkirk, Integer-Sided Triangles with an Angle of 120°, Mathematical Gazette, Vol. 67, No. 442 (Dec., 1983), pp. 251-255.
Eric Weisstein's World of Mathematics, Law of Cosines.
Wikipedia, Triangles with an angle of 120°, Eisenstein triple.

Cf. A357275, A357276, A357277, A357278.

Cf. also A263728, A336750, A335893 (similar with an angle of Pi/3).

for c from 5 to 181 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a,b)=1 and a

Extensions

a(31..33) = 40,51,79 inserted by Georg Fischer, Dec 04 2022

A357275 Smallest side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3.

Original entry on oeis.org

3, 7, 5, 11, 7, 13, 16, 9, 32, 17, 40, 11, 19, 55, 40, 24, 13, 23, 65, 69, 56, 25, 75, 15, 104, 32, 56, 29, 17, 87, 85, 119, 31, 72, 93, 64, 144, 19, 95, 133, 40, 136, 35, 105, 21, 105, 37, 111, 185, 88, 152, 176, 23, 80, 115, 161, 41, 123, 240, 48, 205, 240, 43, 25, 129, 175, 215, 88

Offset: 1

Bernard Schott, Sep 23 2022

nonn

The triples of sides (a,b,c) with a < b < c are in nondecreasing order of largest side c, and if largest sides coincide, then by increasing order of the smallest side. This sequence lists the a's.
For the corresponding primitive triples and miscellaneous properties and references, see A357274.
Solutions a of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
Also, a is generated by integers u, v such that gcd(u,v) = 1 and 0 < v < u, with a = u^2 - v^2.
This sequence is not increasing. For example, a(2) = 7 for triangle with largest side = 13 while a(3) = 5 for triangle with largest side = 19.
Differs from A088514, the first 20 terms are the same then a(21) = 56 while A088514(21) = 25.
A229858 gives all the possible values of the smallest side a, in increasing order without repetition, but for all triples, not necessarily primitive.
All terms of A106505 are values taken by the smallest side a, in increasing order without repetition for primitive triples, but not all the lengths of this side a are present; example: 3 is not in A106505 (see comment in A229849).

			a(2) = a(5) = 7 because 2nd and 5th triple are respectively (7, 8, 13) and (7, 33, 37).

Cf. A357274 (triples), this sequence (smallest side), A357276 (middle side), A357277 (largest side), A357278 (perimeter).

Cf. also A002324, A050931, A088514, A106505, A229849.

for c from 5 to 181 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a, b)=1 and a

Formula

a(n) = A357274(n, 1).

A357276 Middle side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3 = 120 degrees.

Original entry on oeis.org

5, 8, 16, 24, 33, 35, 39, 56, 45, 63, 51, 85, 80, 57, 77, 95, 120, 120, 88, 91, 115, 143, 112, 161, 105, 175, 165, 195, 208, 160, 168, 145, 224, 203, 187, 221, 155, 261, 217, 192, 279, 209, 288, 247, 320, 272, 323, 280, 231, 315, 273, 259, 385, 357, 333, 304, 399, 352, 253, 407, 299, 287, 440

Offset: 1

Bernard Schott, Sep 25 2022

nonn

The triples of sides (a,b,c) with a < b < c are in nondecreasing order of largest side c, and if largest sides coincide, then by increasing order of the smallest side. This sequence lists the b's.
For the corresponding primitive triples and miscellaneous properties and references, see A357274.
Solutions b of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
Also, b is generated by integers u, v such that gcd(u,v) = 1 and 0 < v < u, with b = 2*u*v + v^2.
This sequence is not increasing. For example, a(8) = 56 for triangle with largest side c = 61  while a(9) = 45 for triangle with largest side c = 67.
Differs from A088586, the first 20 terms are the same then a(21) = 115 while A088586(21) = 143.
A229849 gives all the possible values of the middle side b, in increasing order without repetition, for primitive triples, while A229859 gives all the possible values of the middle side b, in increasing order without repetition, but for all triples, not necessarily primitive.

			a(17) = a(18) = 120 since 17th and 18th triples are respectively (13, 120, 127) and (23, 120, 133).

Cf. A357274 (triples), A357275 (smallest side), this sequence (middle side), A357277 (largest side), A357278 (perimeter).

Cf. also A088586, A229849, A229859.

for c from 5 to 500 by 2 do
for a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a,b)=1 and a

A357278 Perimeters of primitive integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees.

Original entry on oeis.org

15, 28, 40, 66, 77, 91, 104, 126, 144, 153, 170, 187, 190, 209, 220, 228, 260, 276, 286, 299, 322, 325, 350, 345, 390, 400, 420, 435, 442, 464, 476, 493, 496, 522, 527, 544, 558, 551, 589, 608, 620, 646, 630, 665, 672, 714, 703, 740, 777, 770, 798, 814, 805

Offset: 1

Bernard Schott, Oct 24 2022

nonn

This sequence lists the sums a+b+c of the triples of sides (a,b,c) of A357274.
Also, sum a+b+c of the solutions of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
For miscellaneous properties, links and references, see A357274.
This sequence is not increasing. For example, a(23) = 350 for triangle with longest side = 163 while a(24) = 345 for triangle with longest side = 169.
Perimeters are in increasing order without repetition in A350045 and perimeters that appear more than once are in A350047.

			(3, 5, 7) is the smallest triple in A357274 with 7^2 = 3^2 + 3*5 + 5^2, so a(1) = 3 + 5 + 7 = 15.

Cf. A350045 (perimeters without repetition), A350047, A357274 (triples), A357275 (smallest side), A357276 (middle side), A357277 (largest side).

for c from 5 to 100 by 2 dofor a from 3 to c-2 do
b := (-a + sqrt(4*c^2-3*a^2))/2;
if b=floor(b) and gcd(a,b)=1 and a

Formula

a(n) = A357274(n, 1) + A357274(n, 2) + A357274(n, 3).

a(n) = A357275(n) + A357276(n) + A357277(n).

cp's OEIS Frontend

A121940 Product of the first n primes of the form 6k+1.

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A357274 List of primitive triples for integer-sided triangles with angles A < B < C and C = 2*Pi/3 = 120 degrees.

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A357275 Smallest side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3.

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A357276 Middle side of integer-sided primitive triangles whose angles satisfy A < B < C = 2*Pi/3 = 120 degrees.

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Maple

A357278 Perimeters of primitive integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees.

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Programs

Maple

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