A010814 -id:A010814 - cp's OEIS frontend

A334718 Largest possible long leg length of a Pythagorean triangle with perimeter A010814(n).

4, 8, 12, 12, 15, 16, 24, 24, 21, 24, 30, 35, 40, 32, 36, 48, 48, 45, 60, 42, 63, 60, 56, 52, 60, 72, 55, 80, 84, 64, 77, 75, 68, 80, 84, 72, 99, 96, 76, 72, 112, 105, 91, 120, 120, 92, 120, 117, 126, 120, 144, 112, 143, 120, 108, 132, 144, 140, 116, 105, 110, 160, 168

Offset: 1

Wesley Ivan Hurt, May 08 2020

nonn

			a(1) = 4; There is one integer-sided right triangle with perimeter A010814(1) = 12, [3,4,5] with long leg 4.
a(2) = 8; There is one integer-sided right triangle with perimeter A010814(2) = 24, [6,8,10] with long leg 8.

Wikipedia, Integer Triangle
Index entries related to Pythagorean Triples.

Cf. A005044, A010814, A334607, A334678, A334717.

A334758 Sum of the perimeters of all Pythagorean triangles with perimeter A010814(n).

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 56, 120, 70, 72, 80, 168, 180, 96, 108, 112, 360, 126, 264, 140, 288, 150, 154, 156, 160, 504, 176, 540, 182, 192, 198, 200, 204, 208, 420, 216, 220, 224, 228, 234, 960, 756, 260, 528, 540, 276, 840, 286, 576, 600, 306, 308, 624, 320, 324, 660, 1008

Offset: 1

Wesley Ivan Hurt, May 10 2020

nonn

			a(1) = 12; There is one integer-sided right triangle with perimeter A010814(1) = 12, [3,4,5].
a(2) = 8; There is one integer-sided right triangle with perimeter A010814(2) = 24, [6,8,10].

Wikipedia, Integer Triangle
Index entries related to Pythagorean Triples.

Cf. A005044, A010814, A334607, A334678, A334717.

A334808 Consider all the Pythagorean triangles with perimeter A010814(n). Then a(n) is the sum of the areas of the squares on all of their sides.

Original entry on oeis.org

50, 200, 338, 450, 578, 800, 1250, 2602, 1682, 1800, 2312, 5188, 6404, 3200, 4050, 5000, 15610, 5618, 13492, 6728, 15650, 8450, 8450, 8450, 9248, 32002, 10658, 36866, 14450, 12800, 14450, 14450, 14450, 15842, 31700, 16200, 20402, 20000, 18050, 18818, 87978, 69164

Offset: 1

Wesley Ivan Hurt, May 12 2020

nonn

			a(1) = 50; there is one Pythagorean triangle with perimeter A010814(1) = 12, [3,4,5]. The sum of the areas of the squares on its sides is 3^2 + 4^2 + 5^2 = 9 + 16 + 25 = 50.
a(2) = 200; there is one Pythagorean triangle with perimeter A010814(2) = 24, [6,8,10]. The sum of the areas of the squares on its sides is 6^2 + 8^2 + 10^2 = 36 + 64 + 100 = 200.

Wikipedia, Integer Triangle
Wikipedia, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A010814.

a(n) = 2 * Sum_{k=1..floor(c(n)/3)} Sum_{i=k..floor((c(n)-k)/2)} sign(floor((i+k)/(c(n)-i-k+1))) * [i^2 + k^2 = (c(n)-i-k)^2] * (c(n)-i-k)^2, where c = A010814. - Wesley Ivan Hurt, May 13 2020

A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

Original entry on oeis.org

1, 1, 1, 3, 2, 2, 2, 2, 2, 1, 1, 2, 7, 3, 1, 1, 3, 3, 3, 3, 2, 2, 3, 12, 4, 1, 1, 1, 1, 4, 4, 4, 4, 2, 1, 1, 2, 4, 15, 5, 2, 1, 1, 2, 5, 5, 3, 5, 5, 2, 2, 2, 2, 5, 9, 9, 6, 2, 1, 1, 1, 1, 2, 6, 6, 6, 6, 3, 1, 1, 1, 1, 3, 6, 28, 7, 2, 2, 1, 1, 2, 2, 7, 7, 7, 7, 3, 2, 1, 1, 2, 3, 7, 12, 12, 8, 3, 1, 2, 2, 1, 3, 8, 8, 8, 8, 8, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Sep 26 2015

Also the rows of both triangles A237270 and A237593 interleaved.
Also, irregular triangle read by rows in which T(n,k) is the area of the k-th region (from left to right in ascending diagonal) of the n-th symmetric set of regions (from the top to the bottom in descending diagonal) in the two-dimensional diagram of the perspective view of the infinite stepped pyramid described in A245092 (see the diagram in the Links section).
The diagram of the symmetric representation of sigma is also the top view of the pyramid, see Links section. For more information about the diagram see also A237593 and A237270.
The number of cubes at the n-th level is also A024916(n), the sum of all divisors of all positive integers <= n.
Note that this pyramid is also a quarter of the pyramid described in A244050. Both pyramids have infinitely many levels.
Odd-indexed rows are also the rows of the irregular triangle A237270.
Even-indexed rows are also the rows of the triangle A237593.
Lengths of the odd-indexed rows are in A237271.
Lengths of the even-indexed rows give 2*A003056.
Row sums of the odd-indexed rows gives A000203, the sum of divisors function.
Row sums of the even-indexed rows give the positive even numbers (see A005843).
Row sums give A245092.
From the front view of the stepped pyramid emerges a geometric pattern which is related to A001227, the number of odd divisors of the positive integers.
The connection with the odd divisors of the positive integers is as follows: A261697 --> A261699 --> A237048 --> A235791 --> A237591 --> A237593 --> A237270 --> this sequence.

			Irregular triangle begins:
  1;
  1, 1;
  3;
  2, 2;
  2, 2;
  2, 1, 1, 2;
  7;
  3, 1, 1, 3;
  3, 3;
  3, 2, 2, 3;
  12;
  4, 1, 1, 1, 1, 4;
  4, 4;
  4, 2, 1, 1, 2, 4;
  15;
  5, 2, 1, 1, 2, 5;
  5, 3, 5;
  5, 2, 2, 2, 2, 5;
  9, 9;
  6, 2, 1, 1, 1, 1, 2, 6;
  6, 6;
  6, 3, 1, 1, 1, 1, 3, 6;
  28;
  7, 2, 2, 1, 1, 2, 2, 7;
  7, 7;
  7, 3, 2, 1, 1, 2, 3, 7;
  12, 12;
  8, 3, 1, 2, 2, 1, 3, 8;
  8, 8, 8;
  8, 3, 2, 1, 1, 1, 1, 2, 3, 8;
  31;
  9, 3, 2, 1, 1, 1, 1, 2, 3, 9;
  ...
Illustration of the odd-indexed rows of triangle as the diagram of the symmetric representation of sigma which is also the top view of the stepped pyramid:
.
   n  A000203    A237270    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
   1     1   =      1      |_| | | | | | | | | | | | | | | |
   2     3   =      3      |_ _|_| | | | | | | | | | | | | |
   3     4   =    2 + 2    |_ _|  _|_| | | | | | | | | | | |
   4     7   =      7      |_ _ _|    _|_| | | | | | | | | |
   5     6   =    3 + 3    |_ _ _|  _|  _ _|_| | | | | | | |
   6    12   =     12      |_ _ _ _|  _| |  _ _|_| | | | | |
   7     8   =    4 + 4    |_ _ _ _| |_ _|_|    _ _|_| | | |
   8    15   =     15      |_ _ _ _ _|  _|     |  _ _ _|_| |
   9    13   =  5 + 3 + 5  |_ _ _ _ _| |      _|_| |  _ _ _|
  10    18   =    9 + 9    |_ _ _ _ _ _|  _ _|    _| |
  11    12   =    6 + 6    |_ _ _ _ _ _| |  _|  _|  _|
  12    28   =     28      |_ _ _ _ _ _ _| |_ _|  _|
  13    14   =    7 + 7    |_ _ _ _ _ _ _| |  _ _|
  14    24   =   12 + 12   |_ _ _ _ _ _ _ _| |
  15    24   =  8 + 8 + 8  |_ _ _ _ _ _ _ _| |
  16    31   =     31      |_ _ _ _ _ _ _ _ _|
  ...
The above diagram arises from a simpler diagram as shown below.
Illustration of the even-indexed rows of triangle as the diagram of the deployed front view of the corner of the stepped pyramid:
.
.                                 A237593
Level                               _ _
1                                 _|1|1|_
2                               _|2 _|_ 2|_
3                             _|2  |1|1|  2|_
4                           _|3   _|1|1|_   3|_
5                         _|3    |2 _|_ 2|    3|_
6                       _|4     _|1|1|1|1|_     4|_
7                     _|4      |2  |1|1|  2|      4|_
8                   _|5       _|2 _|1|1|_ 2|_       5|_
9                 _|5        |2  |2 _|_ 2|  2|        5|_
10              _|6         _|2  |1|1|1|1|  2|_         6|_
11            _|6          |3   _|1|1|1|1|_   3|          6|_
12          _|7           _|2  |2  |1|1|  2|  2|_           7|_
13        _|7            |3    |2 _|1|1|_ 2|    3|            7|_
14      _|8             _|3   _|1|2 _|_ 2|1|_   3|_             8|_
15    _|8              |3    |2  |1|1|1|1|  2|    3|              8|_
16   |9                |3    |2  |1|1|1|1|  2|    3|                9|
...
The number of horizontal line segments in the n-th level in each side of the diagram equals A001227(n), the number of odd divisors of n.
The number of horizontal line segments in the left side of the diagram plus the number of the horizontal line segment in the right side equals A054844(n).
The total number of vertical line segments in the n-th level of the diagram equals A131507(n).
The diagram represents the first 16 levels of the pyramid.
The diagram of the isosceles triangle and the diagram of the top view of the pyramid shows the connection between the partitions into consecutive parts and the sum of divisors function (see also A286000 and A286001). - _Omar E. Pol_, Aug 28 2018
The connection between the isosceles triangle and the stepped pyramid is due to the fact that this object can also be interpreted as a pop-up card. - _Omar E. Pol_, Nov 09 2022

Robert Price, Table of n, a(n) for n = 1..16048 (n=1..412 rows)
Omar E. Pol, An infinite stepped pyramid (A237593, A237270, A262626)
Omar E. Pol, Diagram of the isosceles triangle A237593 before the 90-degree-zig-zag folding (rows: 1..28)
Omar E. Pol, Perspective view of the stepped pyramid (levels: 1..16)
Index entries for sequences related to sigma(n)

Famous sequences that are visible in the stepped pyramid:
Cf. A000040 (prime numbers)......., for the characteristic shape see A346871.
Cf. A000079 (powers of 2)........., for the characteristic shape see A346872.
Cf. A000203 (sum of divisors)....., total area of the terraces in the n-th level.
Cf. A000217 (triangular numbers).., for the characteristic shape see A346873.
Cf. A000225 (Mersenne numbers)...., for a visualization see A346874.
Cf. A000384 (hexagonal numbers)..., for the characteristic shape see A346875.
Cf. A000396 (perfect numbers)....., for the characteristic shape see A346876.
Cf. A000668 (Mersenne primes)....., for a visualization see A346876.
Cf. A001097 (twin primes)........., for a visualization see A346871.
Cf. A001227 (# of odd divisors)..., number of subparts in the n-th level.
Cf. A002378 (oblong numbers)......, for a visualization see A346873.
Cf. A008586 (multiples of 4)......, perimeters of the successive levels.
Cf. A008588 (multiples of 6)......, for the characteristic shape see A224613.
Cf. A013661 (zeta(2))............., (area of the horizontal faces)/(n^2), n -> oo.
Cf. A014105 (second hexagonals)..., for the characteristic shape see A346864.
Cf. A067742 (# of middle divisors), # cells in the main diagonal in n-th level.
Other sequences that are visible in the stepped pyramid: A000096, A001065, A001359, A001747, A002939, A002943, A003056, A004125, A004277, A004526, A005279, A006512, A007606, A007607, A082647, A008438, A008578, A008864, A010814, A014106, A014107, A014132, A014574, A016945, A019434, A024206, A024916, A028552, A028982, A028983, A034856, A038550, A047836, A048050, A052928, A054735, A054844, A062731, A065091, A065475, A071561, A071562, A071904, A092506, A100484, A108605, A139256, A139257, A144396, A152677, A152678, A153485, A155085, A161680, A161983, A162917, A174905, A174973, A175254, A176810, A224880, A235791, A237270, A237271, A237591, A237593, A238005, A238524, A244049, A245092, A259176, A259177, A261348, A278972, A317302, A317303, A317304, A317305, A317307, A319529, A319796, A319801, A319802, A327329, A336305, (and several others).
Apart from zeta(2) other constants that are related to the stepped pyramid are A072691, A353908, A354238.
Cf. A054844, A131507, A196020, A236104, A237048, A239660, A244050, A259179, A261350, A261697, A261699, A262612, A280850, A286000, A286001, A296508.

A005279 Numbers having divisors d, e with d < e < 2*d.

Original entry on oeis.org

6, 12, 15, 18, 20, 24, 28, 30, 35, 36, 40, 42, 45, 48, 54, 56, 60, 63, 66, 70, 72, 75, 77, 78, 80, 84, 88, 90, 91, 96, 99, 100, 102, 104, 105, 108, 110, 112, 114, 117, 120, 126, 130, 132, 135, 138, 140, 143, 144, 150, 153, 154, 156, 160, 162, 165, 168, 170, 174, 175, 176

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

The arithmetic and harmonic means of A046793(n) and a(n) are both integers.
n is in this sequence iff n is a multiple of some term in A020886.
a(n) is also a positive integer v for which there exists a smaller positive integer u such that the contraharmonic mean (uu+vv)/(u+v) is an integer c (in fact, there are two distinct values u giving with v the same c). - Pahikkala Jussi, Dec 14 2008
A174903(a(n)) > 0; complement of A174905. - Reinhard Zumkeller, Apr 01 2010
Also numbers n such that A239657(n) > 0. - Omar E. Pol, Mar 23 2014
Erdős (1948) shows that this sequence has a natural density, so a(n) ~ k*n for some constant k. It can be shown that k < 3.03, and by numerical experiments it seems that k is around 1.8. - Charles R Greathouse IV, Apr 22 2015
Numbers k such that at least one of the parts in the symmetric representation of sigma(k) has width > 1. - Omar E. Pol, Dec 08 2016
Erdős conjectured that the asymptotic density of this sequence is 1. The numbers of terms not exceeding 10^k for k = 1, 2, ... are 1, 32, 392, 4312, 45738, 476153, 4911730, 50359766, 513682915, 5224035310, ... - Amiram Eldar, Jul 21 2020
Numbers with at least one partition into two distinct parts (s,t), sWesley Ivan Hurt, Jan 16 2022
Appears to be the set of numbers x such that there exist numbers y and z satisfying the condition (x^2+y^2)/(x^2+z^2) = (x+y)/(x+z). For example, (15^2+10^2)/(15^2+3^2) = (15+10)/(15+3), so 15 is in the sequence. - Gary Detlefs, Apr 01 2023
From Bob Andriesse, Nov 26 2023: (Start)
Rewriting (x^2+y^2) / (x^2+z^2) = (x+y) / (x+z) as (x^2+y^2) / (x+y) = (x^2+z^2) / (x+z) has the advantage that the values on both sides of the = sign in the given example become integers. A possible sequence with the name: "k's for which r = (k^2+m^2) / (k+m) can be an integer while mA053629(n) and the r's being A009003(n). If (k^2+m^2) / (k+m) = r and m satisfies the divisibility condition, then r-m also does, because (k^2 + (r-m)^2) / (k + (r-m)) = r as well, confirming Pahikkala Jussi's comment about the existence of two distinct values for his u.
The fact that 15 is in the sequence is not so much because (15^2 + 10^2) / (15^2 + 3^2) = 1.3888... = (15+10) / (15+3), as indicated by Gary Detlefs, but rather because (15+10) | (15^2 + 10^2). And since r = (15^2 + 10^2) / (15+10) = 13, the second value that satisfies the divisibility condition is 13-10 = 3, so (15^2 + 3^2) / (15+3) = 13 as well.
Since (k+m)| (k^2 + m^2) is equivalent to (k+m) | 2*k^2 as well as to (k+m) | 2*m^2, both of these alternative divisibility conditions can be used to generate the same sequence too. (End)

R. K. Guy, Unsolved Problems in Number Theory, E3.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Thomas Bloom, The density of integers which have two divisors d_1, d_2 such that d_1 < d_2 < 2*d_1 exists and is equal to 1, Erdős Problems.
Paul Erdős, On the density of some sequences of numbers, Bull. Amer. Math. Soc. 54 (1948), 685--692 MR10,105b; Zbl 32,13 (see Theorem 3).
Paul Erdős, Some unconventional problems in number theory, Journées Arithmétiques de Luminy, Astérisque 61 (1979), p. 73-82.
Paul Erdős, Some unconventional problems in number theory, Mathematics Magazine, Vol. 52, No. 2 (1979), pp. 67-70.
Paul Erdős, On some applications of probability to analysis and number theory, J. London Math. Soc., Vol. 39, No. 1 (1964), pp. 692-696, alternative link.
Planet Math, Integer Contraharmonic Means, Proposition 4.
Planet Math, Contraharmonic proportion
Terence Tao, Erdős problem database, see no. 144.
Robert G. Wilson v, Letter, N.D.

Subsequence of A024619 and hence of A002808.

Cf. A010814, A089341, A020886, A046793, A174903, A174905, A237271, A237593, A239657.

a005279 n = a005279_list !! (n-1)
a005279_list = filter ((> 0) . a174903) [1..]
-- Reinhard Zumkeller, Sep 29 2014

isA005279 := proc(n) local divs,d,e ; divs := numtheory[divisors](n) ; for d from 1 to nops(divs)-1 do for e from d+1 to nops(divs) do if divs[e] < 2*divs[d] then RETURN(true) ; fi ; od: od: RETURN(false) : end; for n from 3 to 300 do if isA005279(n) then printf("%d,",n) ; fi ; od : # R. J. Mathar, Jun 08 2006

aQ[n_] := Select[Partition[Divisors[n], 2, 1], #[[2]] < 2 #[[1]] &] != {}; Select[Range[178], aQ] (* Jayanta Basu, Jun 28 2013 *)

is(n)=my(d=divisors(n));for(i=3,#d,if(d[i]<2*d[i-1],return(1)));0 \\ Charles R Greathouse IV, Apr 22 2015

from sympy import divisors
def is_A005279(n): D=divisors(n)[1:]; return any(e<2*d  for d,e in zip(D, D[1:]))
# M. F. Hasler, Mar 20 2025

a(n) = A010814(n)/2. - Omar E. Pol, Dec 04 2016

A024364 Ordered perimeters of primitive Pythagorean triangles.

Original entry on oeis.org

12, 30, 40, 56, 70, 84, 90, 126, 132, 144, 154, 176, 182, 198, 208, 220, 234, 240, 260, 286, 306, 312, 330, 340, 374, 380, 390, 408, 418, 420, 442, 456, 462, 476, 494, 510, 532, 544, 546, 552, 570, 598, 608, 644, 646, 650, 672, 684, 690, 700, 714, 736, 756

Offset: 1

David W. Wilson

nonn

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives perimeters A+B+C.
k is in this sequence iff A070109(k) > 0. This is a subsequence of A010814.
For the corresponding primitive Pythagorean triples see A103606. - Wolfdieter Lang, Oct 06 2014
Any term in this sequence can be generated by f(m,k) = 2*m*(m+k), where m and k are positive coprime integers and m > 1, k < m, and m and k are not both odd. For example: f(2,1) = 2*2*(2+1) = 12. - Agola Kisira Odero, Apr 29 2016

Ray Chandler, Table of n, a(n) for n = 1..10000 (duplicates removed by Sean A. Irvine)
Leon Bernstein, On primitive Pythagorean triangles with equal perimeters, The Fibonacci Quarterly 27.1 (1989) 2-6 (and the earlier Bernstein paper 20.3 (1982) 227-241, see A024408).
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020886 (semiperimeters: a(n)/2), A024408 (terms with multiplicity > 1).

isA024364 := proc(an) local r::integer,s::integer ; for r from floor((an/4)^(1/2)) to floor((an/2)^(1/2)) do for s from r-1 to 1 by -2 do if 2*r*(r+s) = an and gcd(r,s) < 2 then RETURN(true) ; fi ; if 2*r*(r+s) < an then break ; fi ; od ; od : RETURN(false) ; end : for n from 2 to 400 do if isA024364(n) then printf("%d,",n) ; fi ; od ; # R. J. Mathar, Jun 08 2006

isA024364[an_] := Module[{r, s}, For[r = Floor[(an/4)^(1/2)], r <= Floor[(an/2)^(1/2)], r++, For[s = r - 1, s >= 1, s -= 2, If[2r(r + s) == an && GCD[r, s] < 2, Return[True]]; If[2r(r + s) < an, Break[]]]]; Return[False]];
Select[Range[2, 1000], isA024364] (* Jean-François Alcover, May 24 2024, after R. J. Mathar *)

select( {is_A024364(n)=my(k=valuation(n,2), o=n>>k); k && fordiv(o, r, r^2<<(k-1) >= o && return; r^2< o && gcd(r,o/r)==1 && return(1))}, [1..400]*2) \\ M. F. Hasler, Jul 08 2025

a(n) = 2*A020886(n).

A136003 Primes that are not the sum, minus 1, of a Pythagorean triple.

Original entry on oeis.org

2, 3, 5, 7, 13, 17, 19, 31, 37, 41, 43, 53, 61, 67, 73, 97, 101, 103, 109, 113, 127, 137, 151, 157, 163, 173, 193, 211, 229, 241, 257, 271, 277, 281, 283, 293, 313, 317, 331, 337, 353, 367, 397, 401, 409, 421, 433, 457, 463, 487, 499, 521, 523, 541, 547, 557

Offset: 1

Omar E. Pol, Dec 16 2007

nonn

Primes in A136002.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A009096, A010814, A136002.

q[n_] := PrimeQ[n] && (n == 2 || Module[{d = Divisors[(n+1)/2]}, AllTrue[Range[3, Length[d]], d[[#]] >= 2 * d[[#-1]] &]]); Select[Range[600], q] (* Amiram Eldar, Oct 19 2024 *)

Extended by Ray Chandler, Dec 13 2008

A136002 Numbers that are not the sum, minus 1, of a Pythagorean triple.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 84

Offset: 1

Omar E. Pol, Dec 15 2007

nonn

Numbers that are not in A136000.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A009096, A010814, A136000.

q[n_] := EvenQ[n] || Module[{d = Divisors[(n+1)/2]}, AllTrue[Range[3, Length[d]], d[[#]] >= 2 * d[[#-1]] &]]; Select[Range[100], q] (* Amiram Eldar, Oct 19 2024 *)

Extended by Ray Chandler, Dec 13 2008

A098714 Only one Pythagorean triangle of this perimeter exists.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 56, 70, 72, 80, 96, 108, 112, 126, 140, 150, 154, 156, 160, 176, 182, 192, 198, 200, 204, 208, 216, 220, 224, 228, 234, 260, 276, 286, 306, 308, 320, 324, 340, 348, 350, 352, 364, 372, 374, 378, 380, 384, 392, 400, 416, 418, 442, 444

Offset: 1

Marcus Rezende (marcus(AT)anp.gov.br), Sep 29 2004

nonn

Previous name was : This is the perimeter (n) of square triangles with integer sides and that have only a single solution.

Numbers in A010814 not in A009129. - Hugo Pfoertner, Mar 29 2018

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from Hugo Pfoertner)
Project Euler, Problem 75: Singular integer right triangles.
Index entries related to Pythagorean triples.

Cf. A009129, A010814.

forstep(p=12,444,2,d=0;for(k=1,p-3,for(j=k+1,p-k-1,if(j*j+k*k==(p-j-k)^2,d++)));if(d==1,print1(p,", "))) \\ Hugo Pfoertner, Mar 29 2018

n = a + b + c; c^2=a^2+b^2; a, b, c (sides) and n (perimeter) are integers; for a given "n" there is only a single triple of a, b and c.

More terms from Hugo Pfoertner and Ray Chandler, Oct 27 2004

New name from Hugo Pfoertner, Mar 29 2018

A379374 Irregular triangle read by rows in which row n lists the divisors of n except the divisors "e" described in A005279.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 2, 4, 1, 5, 1, 2, 6, 1, 7, 1, 2, 4, 8, 1, 3, 9, 1, 2, 5, 10, 1, 11, 1, 2, 12, 1, 13, 1, 2, 7, 14, 1, 3, 15, 1, 2, 4, 8, 16, 1, 17, 1, 2, 6, 18, 1, 19, 1, 2, 4, 10, 20, 1, 3, 7, 21, 1, 2, 11, 22, 1, 23, 1, 2, 24, 1, 5, 25, 1, 2, 13, 26, 1, 3, 9, 27

Offset: 1

Omar E. Pol, Dec 21 2024

Observation: the sequence of the number of odd terms in row n coincides with at least the first 10000 terms of A237271.

The observation is true for all numbers. For a proof see A379288. - Hartmut F. W. Hoft, Jan 25 2025

			Triangle begins:
  1;
  1,  2;
  1,  3;
  1,  2,  4;
  1,  5;
  1,  2,  6;
  1,  7;
  1,  2,  4,  8;
  1,  3,  9;
  1,  2,  5, 10;
  1, 11;
  1,  2, 12;
  1, 13;
  1,  2,  7, 14;
  1,  3, 15;
  1,  2,  4,  8, 16;
  1, 17;
  1,  2,  6, 18;
  1, 19;
  1,  2,  4, 10, 20;
  ...

Subsequence of A027750.
Row sums give A379384.
Odd terms give A379288.
Cf. A001227, A005279, A010814, A237271, A237593, A239657, A237271, A379379.

row[n_] := Module[{d = Partition[Divisors[n], 2, 1]}, Join[{1}, Select[d, #[[2]] >= 2*#[[1]] &][[;; , 2]]]]; Table[row[n], {n, 1, 27}] // Flatten (* Amiram Eldar, Dec 22 2024 *)

More terms from Alois P. Heinz, Dec 21 2024

cp's OEIS Frontend

A334718 Largest possible long leg length of a Pythagorean triangle with perimeter A010814(n).

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A334758 Sum of the perimeters of all Pythagorean triangles with perimeter A010814(n).

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A334808 Consider all the Pythagorean triangles with perimeter A010814(n). Then a(n) is the sum of the areas of the squares on all of their sides.

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A262626 Visible parts of the perspective view of the stepped pyramid whose structure essentially arises after the 90-degree-zig-zag folding of the isosceles triangle A237593.

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A005279 Numbers having divisors d, e with d < e < 2*d.

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A024364 Ordered perimeters of primitive Pythagorean triangles.

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A136003 Primes that are not the sum, minus 1, of a Pythagorean triple.

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A136002 Numbers that are not the sum, minus 1, of a Pythagorean triple.

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A098714 Only one Pythagorean triangle of this perimeter exists.

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