A050931 -id:A050931 - cp's OEIS frontend

A002476 Primes of the form 6m + 1.

7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, 127, 139, 151, 157, 163, 181, 193, 199, 211, 223, 229, 241, 271, 277, 283, 307, 313, 331, 337, 349, 367, 373, 379, 397, 409, 421, 433, 439, 457, 463, 487, 499, 523, 541, 547, 571, 577, 601, 607, 613, 619

Offset: 1

N. J. A. Sloane

Equivalently, primes of the form 3m + 1.
Rational primes that decompose in the field Q(sqrt(-3)). - N. J. A. Sloane, Dec 25 2017
Primes p dividing Sum_{k=0..p} binomial(2k, k) - 3 = A006134(p) - 3. - Benoit Cloitre, Feb 08 2003
Primes p such that tau(p) == 2 (mod 3) where tau(x) is the Ramanujan tau function (cf. A000594). - Benoit Cloitre, May 04 2003
Primes of the form x^2 + xy - 2y^2 = (x+2y)(x-y). - N. J. A. Sloane, May 31 2014
Primes of the form x^2 - xy + 7y^2 with x and y nonnegative. - T. D. Noe, May 07 2005
Primes p such that p^2 divides Sum_{m=1..2(p-1)} Sum_{k=1..m} (2k)!/(k!)^2. - Alexander Adamchuk, Jul 04 2006
A006512 larger than 5 (Greater of twin primes) is a subsequence of this. - Jonathan Vos Post, Sep 03 2006
A039701(A049084(a(n))) = A134323(A049084(a(n))) = 1. - Reinhard Zumkeller, Oct 21 2007
Also primes p such that the arithmetic mean of divisors of p^2 is an integer: sigma_1(p^2)/sigma_0(p^2) = C. (A000203(p^2)/A000005(p^2) = C). - Ctibor O. Zizka, Sep 15 2008
Fermat knew that these numbers can also be expressed as x^2 + 3y^2 and are therefore not prime in Z[omega], where omega is a complex cubic root of unity. - Alonso del Arte, Dec 07 2012
Primes of the form x^2 + xy + y^2 with x < y and nonnegative. Also see A007645 which also applies when x=y, adding an initial 3. - Richard R. Forberg, Apr 11 2016
For any term p in this sequence, let k = (p^2 - 1)/6; then A016921(k) = p^2. - Sergey Pavlov, Dec 16 2016; corrected Dec 18 2016
For the decomposition p=x^2+3*y^2, x(n) = A001479(n+1) and y(n) = A001480(n+1). - R. J. Mathar, Apr 16 2024

			Since 6 * 1 + 1 = 7 and 7 is prime, 7 is in the sequence. (Also 7 = 2^2 + 3 * 1^2 = (2 + sqrt(-3))(2 - sqrt(-3)).)
Since 6 * 2 + 1 = 13 and 13 is prime, 13 is in the sequence.
17 is prime but it is of the form 6m - 1 rather than 6m + 1, and is therefore not in the sequence.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
David A. Cox, Primes of the Form x^2 + ny^2. New York: Wiley (1989): 8.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 261.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
C. Banderier, Calcul de (-3/p)
Barry Brent, Finite Field Models of Polynomials Interpolating Fourier Coefficients of Modular Functions for Hecke Groups, Integers (2024) Vol. 24, Art. No. A18. See p. 13.
F. S. Carey, On some cases of the Solutions of the Congruence z^p^(n-1)=1, mod p, Proceedings of the London Mathematical Society, Volume s1-33, Issue 1, November 1900, Pages 294-312.
A. Granville and G. Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 526.
K. G. Reuschle, Tafeln complexer Primzahlen, Königl. Akademie der Wissenschaften, Berlin, 1875, p. 1.
Neville Robbins, On the Infinitude of Primes of the Form 3k+1, Fib. Q., 43,1 (2005), 29-30.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Index to sequences related to decomposition of primes in quadratic fields

Cf. A045331, A242660.
For values of m see A024899. Primes of form 3n - 1 give A003627.
These are the primes arising in A024892, A024899, A034936.
A091178 gives prime index.
Cf. A006512, A007528.
Subsequence of A016921 and of A050931.
Cf. A004611 (multiplicative closure).

Filtered(List([0..110],k->6*k+1),n-> IsPrime(n)); # Muniru A Asiru, Mar 11 2019

a002476 n = a002476_list !! (n-1)
a002476_list = filter ((== 1) . (`mod` 6)) a000040_list
-- Reinhard Zumkeller, Jan 15 2013

(#~ 1&p:) >: 6 * i.1000 NB. Stephen Makdisi, May 01 2018

[n: n in [1..700 by 6] | IsPrime(n)]; // Vincenzo Librandi, Apr 05 2011

a := [ ]: for n from 1 to 400 do if isprime(6*n+1) then a := [ op(a), n ]; fi; od: A002476 := n->a[n];

Select[6*Range[100] + 1, PrimeQ[ # ] &] (* Stefan Steinerberger, Apr 06 2006 *)

select(p->p%3==1,primes(100)) \\ Charles R Greathouse IV, Oct 31 2012

From R. J. Mathar, Apr 03 2011: (Start)
Sum_{n >= 1} 1/a(n)^2 = A175644.
Sum_{n >= 1} 1/a(n)^3 = A175645. (End)
a(n) = 6*A024899(n) + 1. - Zak Seidov, Aug 31 2016
From Vaclav Kotesovec, May 02 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = 1/A175646.
Product_{k>=1} (1 + 1/a(k)^2) = A334481.
Product_{k>=1} (1 - 1/a(k)^3) = A334478.
Product_{k>=1} (1 + 1/a(k)^3) = A334477. (End)
Legendre symbol (-3, a(n)) = +1 and (-3, A007528(n)) = -1, for n >= 1. For prime 3 one sets (-3, 3) = 0. - Wolfdieter Lang, Mar 03 2021

A229858 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of A.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70

Offset: 1

Colin Barker, Oct 06 2013

A229859 gives the values of B, and A050931 gives the values of C.

This sequence contains every integer larger than 8. - Nathaniel Johnston, Oct 06 2013

			12 appears in the sequence because there exists a 120-degree triangle with sides 12, 20 and 28.

Wikipedia, Integer triangle
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (2, -1).

Cf. A050931, A229849, A229859.

\\ Gives values of A not exceeding amax.
\\ e.g. t120a(20) gives [3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
t120a(amax) = {
  v=pt120a(amax);
  s=[];
  for(i=1, #v,
    for(m=1, amax\v[i],
      if(v[i]*m<=amax, s=concat(s, v[i]*m))
    )
  );
  vecsort(s,,8)
}
\\ Gives values of A not exceeding amax in primitive triangles.
\\ e.g. pt120a(20) gives [3, 5, 7, 9, 11, 13, 15, 16, 17, 19]
pt120a(amax) = {
  s=[];
  for(m=1, (amax-1)\2,
    for(n=1, m-1,
      if((m-n)%3!=0 && gcd(m, n)==1,
        a=m*m-n*n;
        b=n*(2*m+n);
        if(a>b, a=b);
        if(a<=amax, s=concat(s, a))
      )
    )
  );
  vecsort(s,,8)
}

a(n) = n+4 for n>4.
a(n) = 2*a(n-1)-a(n-2) for n>6.
G.f.: -x*(x^5-x^4+x^2+x-3) / (x-1)^2.

A232437 Numbers whose square is expressible in only one way as x^2+xy+y^2, with x and y > 0.

Original entry on oeis.org

7, 13, 14, 19, 21, 26, 28, 31, 35, 37, 38, 39, 42, 43, 52, 56, 57, 61, 62, 63, 65, 67, 70, 73, 74, 76, 77, 78, 79, 84, 86, 93, 95, 97, 103, 104, 105, 109, 111, 112, 114, 117, 119, 122, 124, 126, 127, 129, 130, 134, 139, 140, 143, 146, 148, 151, 152, 154, 155, 156, 157, 158, 161

Offset: 1

Jean-Christophe Hervé, Nov 24 2013

nonn

Analog of A084645 for 120-degree angle triangles with integer sides.
Numbers with exactly one prime divisor of the form 6k+1 with multiplicity one.
Primitive elements of A050931.

			a(1) = 7 as 7^2 = 3^2 + 3*5 + 5^2.

Cf. A002476, A050931, A230780, A232436 (subsequence).

Cf. A084645, A232437, A248599, A254063, A254064.

r[k_] := Reduce[x>0 && y>0 && k^2 == x^2 + x y + y^2, {x, y}, Integers];
selQ[k_] := Which[rk = r[k]; rk === False, False, rk[[0]] === And && Length[rk] == 2, False, rk[[0]] === Or && Length[rk] == 2, True, True, False];
Select[Range[1000], selQ] (* Jean-François Alcover, Feb 20 2020 *)

Terms are obtained by the products A230780(k)*A002476(p) for k, p > 0, ordered by increasing values.

A357277 Largest side c of primitive triples, in nondecreasing order, for integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees.

Original entry on oeis.org

7, 13, 19, 31, 37, 43, 49, 61, 67, 73, 79, 91, 91, 97, 103, 109, 127, 133, 133, 139, 151, 157, 163, 169, 181, 193, 199, 211, 217, 217, 223, 229, 241, 247, 247, 259, 259, 271, 277, 283, 301, 301, 307, 313, 331, 337, 343, 349, 361, 367, 373, 379, 397, 403, 403, 409, 421, 427, 427, 433, 439, 457

Offset: 1

Bernard Schott, Oct 01 2022

nonn

For the corresponding primitive triples and miscellaneous properties and references, see A357274.
Solutions c of the Diophantine equation c^2 = a^2 + a*b + b^2 with gcd(a,b) = 1 and a < b.
Also, side c can be generated with integers u, v such that gcd(u,v) = 1 and 0 < v < u, c = u^2 + u*v + v^2.
Some properties:
-> Terms are primes of the form 6k+1, or products of primes of the form 6k+1.
-> The lengths c are in A004611 \ {1} without repetition, in increasing order.
-> Every term appears 2^(k-1) (k>=1) times consecutively.
-> The smallest term that appears 2^(k-1) times is precisely A121940(k): see examples.
-> The terms that appear only once in this sequence are in A133290.
-> The terms are the same as in A335895 but frequency is not the same: when a term appears m times consecutively here, it appears 2m times consecutively in A335895. This is because 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).
Differs from A088513, the first 20 terms are the same then a(21) = 151 while A088513(21) = 157.
A050931 gives all the possible values of the largest side c, in increasing order without repetition, for all triangles with an angle of 120 degrees, but not necessarily primitive.

			c = 7 appears once because A121940(1) = 7 with triple (3,5,7) and 7^2 = 3^2 + 3*5 + 5^2.
c = 91 is the smallest term to appear twice because A121940(2) = 91 with primitive 120-triples (11, 85, 91) and (19, 80, 91).
c = 1729 is the smallest term to appear four times because A121940(3) = 1729 with triples (96, 1679, 1729), (249, 1591, 1729), (656, 1305, 1729), (799, 1185, 1729).

Emrys Read, On integer-sided triangles containing angles of 120° or 60°, Mathematical Gazette 90, July 2006, 299-305.

Cf. A357274 (triples), A357275(smallest side), A357276 (middle side), A357278 (perimeter).

Cf. Also A004611, A050931, A088513, A121940, A133290, A335895.

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

Formula

a(n) = A357274(n, 3).

A229839 Consider all 60-degree triangles with sides A < B < C. The sequence gives the values of C.

Original entry on oeis.org

8, 15, 16, 21, 24, 30, 32, 35, 40, 42, 45, 48, 55, 56, 60, 63, 64, 65, 70, 72, 75, 77, 80, 84, 88, 90, 91, 96, 99, 104, 105, 110, 112, 117, 119, 120, 126, 128, 130, 133, 135, 136, 140, 143, 144, 147, 150, 152, 153, 154, 160, 165, 168, 171, 175, 176, 180, 182

Offset: 1

Colin Barker, Oct 01 2013

nonn

A009005 gives the values of A, and A050931 gives the values of B.

The side n of an equilateral triangle for which a nontrivial integral cevian of length less than n exists, which divides an edge into two integral parts. - Colin Barker, Sep 09 2014

			16 appears in the sequence because there exists a 60-degree triangle with sides 6, 14 and 16.

Wikipedia, Integer triangle
Wikipedia, Cevian

Cf. A009005, A050931, A229838.

Cf. A246918, A246919, A246920.

list={};cmax=182;
Do[If[IntegerQ[Sqrt[e^2-e t+t^2]],AppendTo[list,e]],{e,2,cmax},{t,1,e-1}]
list//DeleteDuplicates (* Herbert Kociemba, Apr 25 2021 *)

\\ Gives values of C not exceeding cmax.
\\ e.g. t60c(60) gives [8, 15, 16, 21, 24, 30, 32, 35, 40, 42, 45, 48, 55, 56, 60]
t60c(cmax) = {
  v=pt60c(cmax);
  s=[];
  for(i=1, #v,
    for(m=1, cmax\v[i],
      if(v[i]*m<=cmax, s=concat(s, v[i]*m))
    )
  );
  vecsort(s,,8)
}
\\ Gives values of C not exceeding cmax in primitive triangles.
\\ e.g. pt60c(115) gives [8, 15, 21, 35, 40, 48, 55, 65, 77, 80, 91, 96, 99, 112]
pt60c(cmax) = {
  s=[];
  for(m=1, ceil(sqrt(cmax+1)),
   for(n=1, m-1,
      if((m-n)%3!=0 && gcd(m, n)==1,
        if(2*m*n+m*m<=cmax, s=concat(s, 2*m*n+m*m))
      )
    )
  );
  vecsort(s,,8)
}

A229859 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of B.

Original entry on oeis.org

5, 8, 10, 15, 16, 20, 24, 25, 30, 32, 33, 35, 39, 40, 45, 48, 50, 51, 55, 56, 57, 60, 63, 64, 65, 66, 70, 72, 75, 77, 78, 80, 85, 88, 90, 91, 95, 96, 99, 100, 102, 104, 105, 110, 112, 114, 115, 117, 120, 125, 126, 128, 130, 132, 135, 136, 140, 143, 144, 145

Offset: 1

Colin Barker, Oct 06 2013

nonn

A229858 gives the values of A, and A050931 gives the values of C.

			20 appears in the sequence because there exists a 120-degree triangle with sides 12, 20 and 28.

Wikipedia, Integer triangle

Cf. A050931, A229849, A229858.

\\ Gives values of B not exceeding bmax.
\\ e.g. t120b(40) gives [5, 8, 10, 15, 16, 20, 24, 25, 30, 32, 33, 35, 39, 40]
t120b(bmax) = {
  v=pt120b(bmax);
  s=[];
  for(i=1, #v,
    for(m=1, bmax\v[i],
      if(v[i]*m<=bmax, s=concat(s, v[i]*m))
    )
  );
  vecsort(s,,8)
}
\\ Gives values of B not exceeding bmax in primitive triangles.
\\ e.g. pt120b(40) gives [5, 8, 16, 24, 33, 35, 39]
pt120b(bmax) = {
  s=[];
  for(m=1, (bmax-1)\2,
    for(n=1, m-1,
      if((m-n)%3!=0 && gcd(m, n)==1,
        a=m*m-n*n;
        b=n*(2*m+n);
        if(a>b, b=a);
        if(b<=bmax, s=concat(s, b))
      )
    )
  );
  vecsort(s,,8)
}

A230780 Positive numbers without a prime factor congruent to 1 (mod 6).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 15, 16, 17, 18, 20, 22, 23, 24, 25, 27, 29, 30, 32, 33, 34, 36, 40, 41, 44, 45, 46, 47, 48, 50, 51, 53, 54, 55, 58, 59, 60, 64, 66, 68, 69, 71, 72, 75, 80, 81, 82, 83, 85, 87, 88, 89, 90, 92, 94, 96, 99, 100, 101, 102, 106, 107

Offset: 1

Jean-Christophe Hervé, Nov 23 2013

nonn

The sequence is closed under multiplication. Primitive elements are 3 and the primes of form 3*k+2.
a(n)^2 is not expressible as x^2+xy+y^2 with x and y positive integers.
Analog of A004144 (nonhypotenuse numbers) for 120-degree angle triangles: a(n) is not the length of the longest side of such a triangle with integer sides.
It might have been natural to include 0 in this sequence. - M. F. Hasler, Mar 04 2018

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 254 terms from Jean-Christophe Hervé)
F. Javier de Vega, An extension of Furstenberg's theorem of the infinitude of primes, arXiv:2003.13378 [math.NT], 2020.
August Lösch, Economics of Location (1954), see pp. 117f.
U. P. Nair, Elementary results on the binary quadratic form a^2+ab+b^2, arXiv:math/0408107 [math.NT], 2004.
Index entries for sequences related to A2 = hexagonal = triangular lattice

Cf. A002476, A005088, complement of A050931.
Cf. A004144 (analog for 4k+1 primes and right triangles).
Cf. A027748.

a230780 n = a230780_list !! (n-1)
a230780_list = filter (all (/= 1) . map (flip mod 6) . a027748_row) [1..]
-- Reinhard Zumkeller, Apr 09 2014

Join[{1}, Select[Range[2, 110], ! MemberQ[Union[Mod[Transpose[ FactorInteger[#]][[1]], 6]], 1] &]] (* T. D. Noe, Nov 24 2013 *)
Join[{1},Select[Range[110],NoneTrue[FactorInteger[#][[All,1]],Mod[#,6] == 1&]&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Feb 03 2019 *)

is_A230780(n)=!setsearch(Set(factor(n)[,1]%6),1) \\ M. F. Hasler, Mar 04 2018

A005088(a(n)) = 0.

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).

A135412 Integers that equal three times the Heronian mean of two positive integers.

Original entry on oeis.org

3, 6, 7, 9, 12, 13, 14, 15, 18, 19, 21, 24, 26, 27, 28, 30, 31, 33, 35, 36, 37, 38, 39, 42, 43, 45, 48, 49, 51, 52, 54, 56, 57, 60, 61, 62, 63, 65, 66, 67, 69, 70, 72, 73, 74, 75, 76, 77, 78, 79, 81, 84, 86, 87, 90, 91, 93, 95, 96, 97, 98, 99, 102, 103, 104, 105, 108, 109, 111

Offset: 1

Pahikkala Jussi, Feb 17 2008

The Heronian mean of two nonnegative real numbers x and y is (x + y + sqrt(xy))/3. Therefore any number n is the Heronian mean of x = 3n and y = 0 (and also of x = n and y = n).

In particular, the sequence contains all numbers n = 3k which equal three times the Heronian mean of k and itself. If the two integers are required to be distinct then most multiples of 3 are no longer in the sequence: see A050931 for the sequence of integers that equal the Heronian mean of two distinct positive integers. Writing x = r^2*s where s is squarefree, the square root is an integer iff y = k^2*s for some integer k, and thus n = s*(r^2 + k^2 + rk). Therefore this sequence consists of the numbers listed in A024614 and their multiples by squarefree s. - M. F. Hasler, Aug 17 2016

			35 is in the sequence since 5 + 20 + sqrt(5*20) = 35.

Planet Math, Heronian mean is between geometric and arithmetic mean
Eric W. Weisstein, Heronian Mean. From MathWorld--A Wolfram Web Resource.
Wikipedia, Heronian mean

Cf. A024614, A050931, A004612.

Edited and definition corrected, following a remark by Robert Israel, by M. F. Hasler, Aug 17 2016

A357302 Numbers k such that k^2 can be represented as x^2 + x*y + y^2 in more ways than for any smaller k.

Original entry on oeis.org

1, 7, 49, 91, 637, 1729, 12103, 53599, 375193, 1983163, 13882141, 85276009, 596932063, 4178524441, 5201836549, 36412855843, 254889990901, 348523048783, 2439661341481, 17077629390367, 25442182561159, 178095277928113, 1246666945496791, 2009932422331561, 14069526956320927

Offset: 1

Hugo Pfoertner, Sep 25 2022

nonn

Apparently the number of grid points t(n) = {1, 2, 3, 5, 8, 14, 23, ...} (A357303) in the reduced representations as described in the examples matches t(n) = A087503(n-3) + 2 for n >= 3, i.e., t(n) = t(n-1) + 3*t(n-2) - 3*t(n-3) for n >= 5. This coincidence persists up to t(15) = 1823, but t(16) = 2553, whereas the recurrence predicts 3281, which is t(17). It seems that all of the terms generated by the recurrence also appear as record numbers of grid points. However, there are other record numbers in between, of which 2553 is the first occurrence.

			The essential information in the complete set of representations of a square a(n)^2 can be extracted by taking into account the symmetries of the triangular lattice. If r is the number of all representations of a(n)^2, then there are t = (r/6 + 1)/2 pairs of triangular oblique coordinates lying in a sector of angular width Pi/6 completely containing the essential information.
a(1) = 1: r = 6 representations of 1^2 are [-1, 0], [-1, 1], [0, -1], [0, 1], [1, -1], [1, 0] reduced: (6/6 + 1)/2 = 1 grid point [1,0].
a(2) = 7: r = 18 representations of 7^2 = 49 are [-8, 5], [-7, 0], [-7, 7], [-5, -3], [-5, 8], [-3, -5], [-3, 8], [0, -7], [0, 7], [3, -8], [3, 5], [5, -8], [5, 3], [7, -7], [7, 0], [8, -5], [8, -3], [8, 3]; reduced: (18/6 + 1)/2 = 2 grid points [7, 0], [8, 3].
After a(2) = 7 there are no squares with more than 18 representations, e.g., r = 18 for 13^2, 14^2, 19^2, 21^2, ..., 42^2, 43^2.
a(3) = 49: r = 30 representations of 49^2 = 2401 are [-56, 21], [-56, 35], [-55, 16], [-55, 39], [-49, 0], [-49, 49], [-39, -16], [-39, 55], [-35, -21], [-35, 56], [-21, -35], [-21, 56], [-16, -39], [-16, 55], [0, -49], [0, 49], [16, -55], [16, 39], [21, -56], [21, 35], [35, -56], [35, 21], [39, -55], [39, 16], [49, -49], [49, 0], [55, -39], [55, -16], [56, -35], [56, -21]; reduced: (30/6 + 1)/2 = 3 grid points [49, 0], [55, 16], [56, 21].
There are no squares with r > 18 between 49 and 90.
a(4) = 91: r = 54 representations of 91^2 = 8281 are [-105,49], [-105,56], ..., [105, -56], [105,-49]; reduced: (54/6 + 1)/2 = 5 grid points [91, 0], [96, 11], [99, 19], [104, 39], [105, 49].

Cf. A002324, A003136, A004016, A050931, A088534, A230655, A357303.

Cf. A087503, A246360.

a357302(upto) = {my (dmax=0);for (k = 1, upto, my (d = #qfbsolve (Qfb(1,1,1), k^2, 3)); if(d > dmax, print1(k,", "); dmax=d))};
a357302(400000)

\\ more efficient using function list_A344473 (see there)
a355703(maxexp10)= {my (sqterms=select(x->issquare(x), list_A344473 (10^(2*maxexp10))), r=0); for (k=1, #sqterms, my (d = #qfbsolve(Qfb(1,1,1),v[k],3)); if (d>r, print1(sqrtint(v[k]),", "); r=d))};
a355703(17)

from itertools import count, islice
from sympy.abc import x,y
from sympy.solvers.diophantine.diophantine import diop_quadratic
def A357302_gen(): # generator of terms
    c = 0
    for k in count(1):
        if (d:=len(diop_quadratic(x*(x+y)+y**2-k**2))) > c:
            yield k
            c = d
A357302_list = print(list(islice(A357302_gen(),6))) # Chai Wah Wu, Sep 26 2022

cp's OEIS Frontend

A002476 Primes of the form 6m + 1.

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A229858 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of A.

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A232437 Numbers whose square is expressible in only one way as x^2+xy+y^2, with x and y > 0.

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A357277 Largest side c of primitive triples, in nondecreasing order, for integer-sided triangles with angles A < B < C = 2*Pi/3 = 120 degrees.

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A229839 Consider all 60-degree triangles with sides A < B < C. The sequence gives the values of C.

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A229859 Consider all 120-degree triangles with sides A < B < C. The sequence gives the values of B.

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A230780 Positive numbers without a prime factor congruent to 1 (mod 6).

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