James R. Buddenhagen - cp's OEIS frontend

A336272 Length of longest side of a primitive square Heron triangle, i.e., a triangle with relatively prime integer sides and area the square of a positive integer.

17, 26, 120, 370, 392, 567, 680, 697, 847, 1066, 1089, 1183, 1233, 1299, 1371, 1448, 1904, 2009, 2169, 2176, 2281, 2307, 2535, 2600, 2619, 2785, 2845, 2993, 3150, 3370, 3825, 3944, 3983, 4035, 4095, 4290, 4706, 4760, 4879, 4905, 5655, 5811, 5835, 6137, 6375, 6570, 6936, 7202, 7913, 7995

Offset: 1

James R. Buddenhagen, Jul 15 2020

nonn

The triangle [a(23)=2535, 2329, 544] with gcd(2329, 544) = 17 is the first square Heron triangle for which the 3 sides [i, j, k] are not pairwise coprime, i.e., max(gcd(i,j), gcd(i,k), gcd(j,k)) > 1, but gcd(i,j,k) = 1. Are there more square Heron triangles with this property? - Hugo Pfoertner, Jul 18 2020
There are other square Heron triangles with this property, e.g. [a(31)=3825, 2704, 1921] with gcd(1921, 3825) = 17; [a(??)=41460721, 38639097, 17536520] with gcd(38639097, 17536520) = 41; [a(??)=153915025, 139641489, 25224736] with gcd(25224736, 153915025) = 17; and [a(??)=4325561361, 3459908000, 1430190961] with gcd(3459908000, 1430190961) = 73. - James R. Buddenhagen, Jul 20 2020
Terms are given with multiplicity, e.g. if there are two primitive square Heron triangles with equal longest sides, that longest side is listed as a term of the sequence twice (this is very rare). - James R. Buddenhagen, Jul 21 2020

			17 is in the sequence because the triangle with sides [17, 10, 9] has longest side 17 and area 6^2, the square of a positive integer; 26 is in the sequence because the triangle with sides [26, 25, 3] has longest side 26 and has area 6^2, the square of a positive integer.
Triangles with sides [a, b, c] corresponding to the first 8 terms of this sequence are:  [17, 10, 9], [26, 25, 3], [120, 113, 17], [370, 357, 41], [392, 353, 255], [567, 424, 305], [680, 441, 337], [697, 657, 104].

Hugo Pfoertner, Table of n, a(n) for n = 1..79
Sascha Kurz, On the generation of Heronian triangles, arXiv:1401.6150 [math.NT], 11 Jan 2014.
Sascha Kurz, On the generation of Heronian triangles, Serdica Journal of Computing Vol. 2 (2008), Pages 181-196.
Hugo Pfoertner, List of triangle sides, 20000 > i > j > k.

Cf. A046128, A046129, A046130, A046131, A055592, A055593, A055594, A055595.

# find all square Heron triangles whose longest side is between small and big
small:=1: big:=700:
A336272:=[]:triangles:=[]:
areasq16:=(a+b+c)*(a+b-c)*(a-b+c)*(-a+b+c):
# a>=b>=c
for a from small to big do:
  for b from ceil((a+1)/2) to a do:
    for c from a-b+1 to b do:
      if issqr(areasq16) and issqr(sqrt(areasq16)) and igcd(a,b,c)=1 then
        A336272:=[op(A336272),a]:
        triangles:=[op(triangles),[a,b,c]]:
      end if:
    od:
  od:
od: A336272;triangles;

for(a=1,1200,for(b=ceil((a+1)/2),a,for(c=a-b+1,b,if(gcd([a,b,c])==1,if(ispower((a+b+c)*(a+b-c)*(a-b+c)*(b+c-a),4),print1(a,", ")))))) \\ Hugo Pfoertner, Jul 18 2020

a(42)-a(50) from Hugo Pfoertner, Jul 18 2020

A334294 Numbers k such that 70k^2 + 70k - 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 17, 20, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 36, 37, 39, 40, 41, 43, 44, 45, 46, 56, 57, 58, 59, 60, 61, 63, 64, 65, 66, 67, 68, 70, 71, 74, 76, 77, 78, 79, 80, 81, 82, 87, 88, 90, 93, 96, 97, 100

Offset: 1

James R. Buddenhagen, Apr 21 2020

nonn

Among quadratic polynomials in k of the form a*k^2 + a*k - 1 the value a=70 gives the most primes for any a in the range 1<=a<=300, at least up to k=40000. Here a and k are positive integers. Other "good" values of a are a=250, a=99, and a=19.

			For k=1, 70*k^2 + 70*k - 1 = 70*1^2 + 70*1 - 1 = 139, which is prime, so 1 is in the sequence.

N. Boston et M. L.Greenwood, Quadratics representing primes, Amer. Math. Monthly 102:7 (1995), 595-599.
François Dress and Michel Olivier, Polynômes prenant des valeurs premières, Experimental Mathematics, Volume 8, Issue 4 (1999), 319-338.
G. W. Fung and H. C. Williams, Quadratic polynomials which have a high density of prime values, Math. Comput. 55(191) (1990), 345-353.
R. A. Mollin, Prime-Producing Quadratics, The American Mathematical Monthly, Vol. 104, No. 6 (Jun. - Jul., 1997), pp. 529-544.

Cf. A001912, A002384, A005574, A027861, A028870, A056900.

Cf. A067201, A088572, A089623, A091199, A271980.

a:=proc(n) if isprime(70*n^2+70*n-1) then n else NULL end if end proc;
seq(a(n),n=1..100);

Select[Range@ 100, PrimeQ[70 #^2 + 70 # - 1] &] (* Michael De Vlieger, May 26 2020 *)

A330563 Primes in A330562.

Original entry on oeis.org

101, 103, 109, 401, 601, 701, 1009, 1021, 1033, 1051, 1063, 1201, 1303, 1801, 2011, 2017, 2081, 2801, 3001, 3061, 3301, 4001, 4021, 4201, 4801, 5011, 5021, 5051, 5081, 5101, 5501, 5801, 6301

Offset: 1

James R. Buddenhagen, Dec 31 2019

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A330562.

filter:= proc(k) local L;
  L:= convert(convert(k,base,10),set);
  andmap(d -> member(k mod d, L), L minus {0}) and isprime(k)
end proc:
select(filter, [seq(i,i=101..10000,2)]); # Robert Israel, Dec 23 2024

A300473 Numbers k with the property that k^2 + 21k + 1 is prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 73, 77, 78, 79, 80, 81, 82, 84, 85, 86, 87, 89, 91, 97, 100

Offset: 1

James R. Buddenhagen, Mar 06 2018

nonn

The quadratic polynomial p(k) = k^2 + 21*k + 1 is not a prime-generating polynomial in the sense of Eric Weisstein's World of Mathematics (see link) because p(0) is not prime.

However p(k) is prime for the first 17 positive integral values of k and among polynomials of the form k^2 + j*k + 1, the present polynomial (j = 21) generates more primes than any polynomial of that form for any positive integral j < 231, at least for positive integers, k, in the range 0 < k < 10^6.

			17 is in the sequence because 17^2 + 21 * 17 + 1 = 647 is prime.
18 is not in the sequence because 18^2 + 21 * 18 + 1 = 703 = 19 * 37.

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Prime-Generating Polynomial

Cf. A292509, A050267, A050268, A005846, A007635, A007641, A048988.

select(k-> isprime(k^2+21*k+1), [$1..100])

Select[Range[100], PrimeQ[#^2 + 21# + 1] &] (* Alonso del Arte, Mar 06 2018 *)

isok(k) = isprime(k^2+21*k+1); \\ Altug Alkan, Mar 07 2018

A273049 Palindromic primes with exactly three nonzero digits.

Original entry on oeis.org

131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, 10301, 10501, 10601, 30103, 30203, 30403, 30703, 30803, 70207, 70507, 70607, 90709, 1003001, 1008001, 3001003, 3002003, 3007003, 9002009, 100030001, 100050001, 100060001, 300020003

Offset: 1

James R. Buddenhagen, May 13 2016

James R. Buddenhagen and Giovanni Resta, Table of n, a(n) for n = 1..148 (terms < 10^1000, first 70 terms from J. R. Buddenhagen)
Giovanni Resta, Compact representation of the terms up to 10^20000

Subsequence of A002385.

[p: p in PrimesUpTo(100000000) | #[d: d in t[1..#t]| d ne 0] eq 3 and p eq Seqint(Reverse(t)) where t is Intseq(p)]; // Bruno Berselli, May 16 2016

Reap[ Do[ If[PrimeQ[p = e*(1+10^(2*n)) + m*10^n], Sow@p], {n, 99}, {e, {1, 3, 7, 9}}, {m, 9}]][[2, 1]] (* Giovanni Resta, May 14 2016 *)

A264783 Primes that do not divide 2^m - 2293 for any positive integer m.

Original entry on oeis.org

2, 31, 41, 43, 71, 73, 89, 103, 113, 127, 151, 157, 223, 229, 233, 241, 251, 257, 263, 277, 311, 313, 331, 337, 353, 359, 397, 409, 431, 449, 487, 499, 521, 571, 577, 599, 607, 631, 641, 643, 647, 673, 683, 691, 719, 727, 739, 769, 809, 811, 839, 857, 863, 881, 911, 919, 929, 937, 953, 971, 983, 1009, 1013, 1021, 1031, 1033, 1049, 1051, 1063, 1069, 1087, 1093, 1097, 1103, 1151, 1153, 1163, 1181, 1193, 1201, 1217, 1223, 1231, 1249, 1279, 1289, 1297, 1321, 1327, 1367, 1399, 1409, 1423, 1429, 1433, 1439, 1447

Offset: 1

James R. Buddenhagen, Nov 24 2015

nonn

Robert Israel, Table of n, a(n) for n = 1..10000
See some related discussion here: http://math.stackexchange.com/questions/1529814/2i-2293-is-always-composite

select(t -> isprime(t) and numtheory:-mlog(2293,2,t) = FAIL, [2,seq(t,t=3..2000,2)]); # Robert Israel, Nov 25 2015

A263718 n such that (1+2*7^n)/3 is prime.

Original entry on oeis.org

1, 3, 4, 15, 24, 58, 168, 234, 324, 375, 3327, 5788, 8344, 15039, 59116, 62620

Offset: 1

James R. Buddenhagen, Oct 24 2015

nonn

The numbers corresponding to a(11)-a(16) are only (extra-strong BPSW) probable primes. - Charles R Greathouse IV, Nov 10 2015

Select[Range@ 2000, PrimeQ[(1 + 2 7^#)/3] &] (* Michael De Vlieger, Nov 06 2015 *)

is(n)=ispseudoprime((1+2*7^n)/3) \\ Charles R Greathouse IV, Nov 06 2015

a(11)-a(14) from Charles R Greathouse IV, Nov 06 2015

a(15)-a(16) from Charles R Greathouse IV, Nov 09 2015

A231101 a(n) = 3*a(n-3) + a(n-2), a(0)=3, a(1)=0, a(2)=2.

Original entry on oeis.org

3, 0, 2, 9, 2, 15, 29, 21, 74, 108, 137, 330, 461, 741, 1451, 2124, 3674, 6477, 10046, 17499, 29477, 47637, 81974, 136068, 224885, 381990, 633089, 1056645, 1779059, 2955912, 4948994

Offset: 0

James R. Buddenhagen, Nov 05 2013

a(n) = r^n+s^n+t^n, where r,s,t are the roots of x^3-x-3.
If p is prime then p divides a(p).
Both this and the Perrin sequence are linear recurrences with a(n) depending on a(n-3) and a(n-2) but not on a(n-1), with the same initial conditions; both are sums of powers of roots of a cubic: Perrin: a(n) = r^n+s^n+t^n with r,s,t roots of x^3-x-1 this seq: a(n) = r^n+s^n+t^n with r,s,t roots of x^3-x-3.  See crossrefs.

Index entries for linear recurrences with constant coefficients, signature (0,1,3).

Cf. A001608, A072328.

a:=proc(n) option remember:
if n=0 then 3 elif n=1 then 0 elif n=2 then 2 else 3*a(n-3)+a(n-2) end if end proc:
bign:=30:
seq(a(n),n=0..bign);

CoefficientList[Series[(x^2 - 3)/(3*x^3 + x^2 - 1), {x, 0, 50}], x] (* Wesley Ivan Hurt, May 26 2024 *)

a(n) = 3*a(n-3)+a(n-2), a(0)=3, a(1)=0, a(2)=2.
a(n) = r^n+s^n+t^n, where r,s,t are the roots of x^3-x-3.
G.f.: (x^2-3)/(3*x^3+x^2-1).

A206351 a(n) = 7*a(n-1) - a(n-2) - 4 with a(1)=1, a(2)=3.

Original entry on oeis.org

1, 3, 16, 105, 715, 4896, 33553, 229971, 1576240, 10803705, 74049691, 507544128, 3478759201, 23843770275, 163427632720, 1120149658761, 7677619978603, 52623190191456, 360684711361585, 2472169789339635

Offset: 1

James R. Buddenhagen, Feb 06 2012

A Pell sequence related to Heronian triangles (rational triangles), see A206334. The connection is this: consider the problem of finding triangles with area a positive integer n, and with sides (a, b, n) where a, b are rational. Note that n is both the area and one side. For many values of n this is not possible, and the sequence of such numbers n is quite erratic (see A206334). Nonetheless, each term in this sequence is such a value of n. For example, for n = 105 you can take the other two sides, a and b, to be 10817/104, and 233/104 and the area will equal n, i.e., 105.

			G.f. = x + 3*x^2 + 16*x^3 + 105*x^4 + 715*x^5 + 4896*x^6 + 33553*x^7 + ... - _Michael Somos_, Jun 26 2018

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Pridon Davlianidze, Problem B-1279, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 58, No. 4 (2020), p. 368; An Unusual Generalization, Solution to Problem B-1279 by J. N. Senadheera, ibid., Vol. 59, No. 4 (2021), pp. 370-371.
Index entries for linear recurrences with constant coefficients, signature (8,-8,1).

Cf. A000045, A003482, A089508, A094214.

Subsequence of A206334.

a206351 n = a206351_list !! (n-1)
a206351_list = 1 : 3 : map (subtract 4)
               (zipWith (-) (map (* 7) (tail a206351_list)) a206351_list)
-- Reinhard Zumkeller, Feb 08 2012

[Fibonacci(2*n)*Fibonacci(2*n-3): n in [1..30]]; // G. C. Greubel, Aug 12 2018

genZ := proc(n)
local start;
option remember;
    start := [1, 3];
    if n < 3 then start[n]
    else 7*genZ(n - 1) - genZ(n - 2) - 4
    end if
end proc:
seq(genZ(n),n=1..20);

LinearRecurrence[{8, -8, 1}, {1, 3, 16}, 50] (* Charles R Greathouse IV, Feb 07 2012 *)
RecurrenceTable[{a[1] == 1, a[2] == 3, a[n] == 7 a[n - 1] - a[n - 2] - 4}, a, {n, 20}] (* Bruno Berselli, Feb 07 2012 *)
a[ n_] := Fibonacci[2 n] Fibonacci[2 n - 3]; (* Michael Somos, Jun 26 2018 *)
nxt[{a_,b_}]:={b,7b-a-4}; NestList[nxt,{1,3},20][[;;,1]] (* Harvey P. Dale, Aug 29 2024 *)

Vec((1-5*x)/(1-8*x+8*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Feb 07 2012

{a(n) = fibonacci(2*n) * fibonacci(2*n - 3)}; /* Michael Somos, Jun 26 2018 */

From Bruno Berselli, Feb 07 2012: (Start)
G.f.: x*(1-5*x)/(1-8*x+8*x^2-x^3).
a(n) = A081018(n-1) + 1. (End)
a(n) = -A003482(-n) = Fibonacci(2*n)*Fibonacci(2*n-3). - Michael Somos, Jun 26 2018
a(n) = A089508(n-1) + 2 for n>1. - Bruno Berselli, Jun 20 2019 [Formula found by Umberto Cerruti]
Product_{n>=2} (1 - 1/a(n)) = 1/phi (A094214) (Davlianidze, 2020). - Amiram Eldar, Nov 30 2021
a(n) = (Fibonacci(2*n-2) + 1/Lucas(2*n-2))*(Fibonacci(2*n-1) + 1/Lucas(2*n-1)). - Peter Bala, Sep 03 2022

A206334 Numbers n such that there is a triangle with area n, side n, and the other two sides rational.

Original entry on oeis.org

3, 5, 7, 10, 12, 15, 16, 18, 19, 23, 25, 26, 27, 28, 29, 30, 33, 34, 36, 38, 39, 40, 41, 42, 43, 44, 46, 47, 51, 52, 55, 57, 58, 59, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 80, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 95, 96, 97, 103, 104, 105, 106, 107, 109, 115, 119, 122, 123, 124, 125, 126

Offset: 1

James R. Buddenhagen, Feb 06 2012

nonn

n>3 is in the sequence just in case the elliptic curve y^2 = 4*x^4 + (n^2+8)*x^2 + 4 has positive rank.  Note that (0,2) is on that curve.
n is in the sequence just in case there are positive rational numbers x,y such that x*y>1 and x - 1/x + y - 1/y = n.
The triangle whose sides are [(4*k^6+8*k^5+8*k^4+4*k^3+2*k^2+2*k+1)/((k+1)*k*(2*k^2+2*k+1)), (4*k^6+16*k^5+28*k^4+28*k^3+18*k^2+6*k+1)/((k+1)*k*(2*k^2+2*k+1)), 4*k^2+4*k+4] has area equal to its third side.  Hence, starting with the second term, A112087 is a subsequence of the present sequence.
The triangle whose sides are [(k^6+2*k^4+k^2+1)/(k*(k^2+1)), (k^4+3*k^2+1)/(k*(k^2+1)), (k^2+2)*k] has area equal to its third side. Hence, starting with the first positive term, A054602 is a subsequence of the present sequence. [This subsequence found by Dragan K, see second link, below.]
The triangle whose sides are [(k^8+6*k^6+13*k^4+13*k^2+4)/(k*(k^2+2)*(k^2+1)), (k^6+3*k^4+5*k^2+4)/(k*(k^2+2)*(k^2+1)), k*(k^2+4)] has area equal to its third side. Hence A155965 is a subsequence of the present sequence.

			5 is in the sequence because the triangle with sides (37/6, 13/6, 5) has area 5, one side 5, and the other two sides rational.

James R. Buddenhagen, Table of triangles up to n = 145
Dragan K and Rita the dog, Question and answer [broken link]
Ian Connell, APECS elliptic curve software (which runs under old versions of Maple).
Eric Weisstein's World of Mathematics, Heronian Triangle.

Cf. A112087, A054602, A155965, and A206351 (subsequences, see comments).

cp's OEIS Frontend

User: James R. Buddenhagen

A336272 Length of longest side of a primitive square Heron triangle, i.e., a triangle with relatively prime integer sides and area the square of a positive integer.

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A334294 Numbers k such that 70*k^2 + 70*k - 1 is prime.

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Mathematica

A330563 Primes in A330562.

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Maple

A300473 Numbers k with the property that k^2 + 21k + 1 is prime.

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Maple

Mathematica

PARI

A273049 Palindromic primes with exactly three nonzero digits.

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Magma

Mathematica

A264783 Primes that do not divide 2^m - 2293 for any positive integer m.

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Maple

A263718 n such that (1+2*7^n)/3 is prime.

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Extensions

A231101 a(n) = 3*a(n-3) + a(n-2), a(0)=3, a(1)=0, a(2)=2.

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A334294 Numbers k such that 70k^2 + 70k - 1 is prime.