A009005 -id:A009005 - cp's OEIS frontend

A334677 Number of integer-sided triangles with nonintegral area and perimeter A009005(n).

1, 1, 1, 2, 1, 3, 2, 4, 2, 5, 4, 7, 4, 8, 6, 10, 8, 12, 10, 14, 11, 16, 14, 19, 16, 21, 18, 24, 19, 27, 24, 30, 23, 33, 30, 37, 32, 40, 35, 44, 39, 48, 44, 52, 45, 56, 50, 61, 56, 65, 57, 70, 64, 75, 70, 80, 72, 85, 80, 91, 80, 96, 90, 102, 95, 108, 100, 114, 103, 120, 114

Offset: 1

Wesley Ivan Hurt, May 08 2020

nonn

			a(1) = 1; There is one integer-sided triangle with perimeter A009005(1) = 3 whose area is not an integer, [1,1,1] with area sqrt(3)/4.
a(2) = 1; There is one integer-sided triangle with perimeter A009005(2) = 5 whose area is not an integer, [1,2,2] with area sqrt(15)/4.

Wikipedia, Integer Triangle

Cf. A005044, A009005, A051516, A051518.

A024352 Numbers which are the difference of two positive squares, c^2 - b^2 with 1 <= b < c.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32, 33, 35, 36, 37, 39, 40, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 88, 89, 91, 92, 93, 95, 96

Offset: 1

David W. Wilson

These are the solutions to the equation x^2 + xy = n where y mod 2 = 0, y is positive and x is any positive integer. - Andrew S. Plewe, Oct 19 2007
Ordered different terms of A120070 = 3, 8, 5, 15, 12, 7, ... (which contains two 15's, two 40's, and two 48's). Complement: A139544. (See A139491.) - Paul Curtz, Sep 01 2009
A024359(a(n)) > 0. - Reinhard Zumkeller, Nov 09 2012
If a(n) mod 6 = 3, n > 1, then a(n) = c^2 - f(a(n))^2 where f(n) = (floor(4*n/3) - 3 - n)/2. For example, 171 = 30^2 - 27^2 and f(171) = 27. - Gary Detlefs, Jul 15 2014

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Ron Knott, Pythagorean Triples and Online Calculators
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Same as A042965 except for initial terms. - Michael Somos, Jun 08 2000
Different from A020884.
Cf. A009005, A020884, A120070, A139544, A139491.

a024352 n = a024352_list !! (n-1)
a024352_list = 3 : drop 4 a042965_list
-- Reinhard Zumkeller, Nov 09 2012

[3] cat [4 +Floor((4*n-3)/3): n in [2..100]]; // G. C. Greubel, Apr 22 2023

Union[Flatten[Table[Select[Table[b^2 - c^2, {c, b-1}], # < 100 &], {b, 100}]]] (* Robert G. Wilson v, Jun 05 2004 *)
LinearRecurrence[{1,0,1,-1},{3,5,7,8,9},70] (* Harvey P. Dale, Dec 20 2021 *)

is(n)=(n%4!=2 && n>4) || n==3 \\ Charles R Greathouse IV, May 31 2013

def A024352(n): return 3 if n==1 else 3+(n<<2)//3 # Chai Wah Wu, Feb 10 2025

def A024352(n): return 4 + ((4*n-3)//3) - int(n==1)
[A024352(n) for n in range(1,101)] # G. C. Greubel, Apr 22 2023

Consists of all positive integers except 1, 4 and numbers == 2 (mod 4).
a(n) = a(n-3) + 4, n > 4.
G.f.: (3 + 2*x + 2*x^2 - 2*x^3 - x^4)/(1 - x - x^3 + x^4). - Ralf Stephan, before May 13 2008
a(n) = a(n-1) + a(n-3) - a(n-4), for n > 5. - Ant King, Oct 03 2011
a(n) = 4 + floor((4*n-3)/3), n > 1. - Gary Detlefs, Jul 15 2014

Edited by N. J. A. Sloane, Sep 19 2008

A009056 Numbers >= 3.

Original entry on oeis.org

3, 4, 5, 6, 7, 8, 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, 71, 72, 73, 74, 75, 76, 77

Offset: 1

David W. Wilson

Same as Pisot sequences E(3,4), P(3,4), T(3,4). See A008776 for definitions of Pisot sequences.
Non-Fermat-exponents, positive integers n such that there are no solutions in positive integers of the equation a^n + b^n = c^n. - Tanya Khovanova, Jul 09 2011
Sums of twin primes. - Charles R Greathouse IV, Jun 21 2012

Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A009005, A020739, A055998.

Range[3,80] (* Harvey P. Dale, Sep 26 2017 *)

a(n)=n+2 \\ Charles R Greathouse IV, Jun 21 2012

a(n) = n + 2.
From R. J. Mathar, May 26 2008: (Start)
O.g.f.: x*(3-2*x)/(1-x)^2.
a(n) = A009005(n-1), n > 2. (End)
From Elmo R. Oliveira, Oct 31 2024: (Start)
E.g.f.: exp(x)*(x + 2) - 2.
a(n) = 2*a(n-1) - a(n-2) for n > 2.
a(n) = A055998(n) - A055998(n-1) = A020739(n-1)/2. (End)

A057098 Shortest side of a Pythagorean triangle (ordered by the product of the sides).

Original entry on oeis.org

3, 6, 5, 9, 8, 12, 7, 10, 15, 20, 18, 9, 12, 16, 21, 15, 24, 14, 11, 27, 20, 24, 30, 16, 28, 33, 13, 40, 25, 36, 21, 18, 33, 24, 32, 39, 42, 30, 15, 48, 20, 45, 36, 48, 40, 35, 28, 39, 51, 22, 60, 54, 17, 27, 40, 57, 36, 48, 65, 60, 24, 32, 35, 56, 63, 45, 60, 19, 66, 44, 56

Offset: 1

Henry Bottomley, Aug 01 2000

nonn

			a(1)=3 since 3*4*5=60 is smallest possible positive product

Cf. A009004, A009005, A046083, A057096.

maxShortLeg = 66; terms = 71;
r[a_] := {a, b, c} /. {ToRules[Reduce[a <= b < c && a^2+b^2 == c^2, {b, c}, Integers]]};
abc = r /@ Complement[Range[maxShortLeg], {1, 2, 4}] // Flatten[#, 1]&;
SortBy[abc, Times @@ # &][[;; terms, 1]] (* Jean-François Alcover, Nov 21 2019 *)

a(n) =A057096(n)/(A057099(n)*A057100(n)) =sqrt(A057100(n)^2-A057099(n)^2)

A242963 Numbers n such that A242962(n) = sigma(n) = A000203(n).

Original entry on oeis.org

5, 7, 8, 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, 71

Offset: 1

Jaroslav Krizek, May 29 2014

nonn

A242962(n) = (n*(n+1)/2) mod antisigma(n) = A000217(n) mod A024816(n).
Union of number 5 and numbers >= 7.
Conjecture: this sequence lists all the positive integers n such that, for some integer k, (sin(k*Pi/n))^2 is irrational. - Lorenzo Sauras Altuzarra, Jan 27 2020

Michael De Vlieger, Table of n, a(n) for n = 1..14995

Cf. A000203, A000217, A002961, A024816, A242962, A243117, A243118, A217290, A009005.

[n: n in [3..100000] | ((n*(n+1)div 2) mod (n*(n+1)div 2-SumOfDivisors(n))) eq (SumOfDivisors(n))]

Select[Range[3, 71], DivisorSigma[1, #] == Mod[PolygonalNumber@ #, Total@ Complement[Range@ #, Divisors@ #]] &] (* Michael De Vlieger, Jan 28 2020 *)

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)
}

A263772 Perimeters of integer-sided scalene triangles.

Original entry on oeis.org

9, 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, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Rick L. Shepherd, Oct 27 2015

All natural numbers larger than 8 except 10.
Equivalently, numbers n that can be partitioned into three distinct parts a, b, and c, where a + b > c, a + c > b, and b + c > a (or, without loss of generality, into (a, b, c) with a < b < c < a + b). A subsequence of A009005. The unique terms in A107576.
For k > 2, (k-1, k, k+1) gives perimeter 3k and (k-1, k+1, k+2) gives perimeter 3k + 2. For k > 3, the scalene triangle (k-1, k, k+2) has perimeter 3k + 1.

			The integer-sided scalene triangle of least perimeter has sides of lengths 2, 3, and 4, so a(1) = 2 + 3 + 4 = 9.

Cf. A005044, A009005, A107572, A107576.

vector(100, n, if(n==1, 9, n+9)) \\ Altug Alkan, Oct 28 2015

a(n) = n + 9 for n > 1.

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A334677 Number of integer-sided triangles with nonintegral area and perimeter A009005(n).

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A024352 Numbers which are the difference of two positive squares, c^2 - b^2 with 1 <= b < c.

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A009056 Numbers >= 3.

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A057098 Shortest side of a Pythagorean triangle (ordered by the product of the sides).

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A242963 Numbers n such that A242962(n) = sigma(n) = A000203(n).

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Magma

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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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Mathematica

PARI

A263772 Perimeters of integer-sided scalene triangles.

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PARI

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