A008846 -id:A008846 - cp's OEIS frontend

A230486 Numbers n such that n^n is representable as the sum of two nonzero squares.

5, 10, 13, 17, 20, 25, 26, 29, 30, 34, 37, 40, 41, 50, 52, 53, 58, 60, 61, 65, 68, 70, 73, 74, 78, 80, 82, 85, 89, 90, 97, 100, 101, 102, 104, 106, 109, 110, 113, 116, 120, 122, 125, 130, 136, 137, 140, 145, 146, 148, 149, 150, 156, 157, 160, 164, 169, 170

Offset: 1

Alex Ratushnyak, Oct 20 2013

nonn

If n is even, then n must have a prime factor of the form 4k+1. If n is odd, then all prime factors must be of the form 4k+1. - T. D. Noe, Oct 21 2013

The above is also a sufficient condition: the sequence consists exactly in even multiples of Pythagorean primes A002144, and products of such primes (A008846). - M. F. Hasler, Sep 02 2018

			5^5 = 55^2 + 10^2.
10^10 = 99712^2 + 7584^2.
13^13 = 17106843^2 + 3198598^2.
17^17 = 28735037644^2 + 1240110271^2.

G. H. Hardy and E. M. Wright, Theory of Numbers, Oxford, Sixth Edition, 2008, p. 395.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A000312 (n^n), A004431, A132777.
A subsequence of A000404 (numbers that are the sum of 2 nonzero squares).
Sequence A002144 (primes of the form 4k + 1) and A008846 (products of such primes) are subsequences.

t = {}; Do[f = FactorInteger[n]; p = Transpose[f][[1]]; If[EvenQ[n], If[MemberQ[Mod[p, 4], 1], AppendTo[t, n]], If[Union[Mod[p, 4]] == {1}, AppendTo[t, n]]], {n, 2, 200}]; t (* T. D. Noe, Oct 21 2013 *)

select( is_A230486(n)={(n=factor(n)[,1]%4) && if(n[1]==2, Set(n)[1]==1, Set(n)==[1])}, [1..200]) \\ M. F. Hasler, Sep 02 2018

from itertools import count, islice
from sympy import primefactors
def A230486_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:all(p&3==1 for p in primefactors(n)) if n&1 else any(p&3==1 for p in primefactors(n)),count(max(startvalue,2)))
A230486_list = list(islice(A230486_gen(),20)) # Chai Wah Wu, May 15 2023

A230486 = { n | A000312(n) is in A000404 } = A004277*A002144 U A008846. - M. F. Hasler, Sep 02 2018

Extended by T. D. Noe, Oct 21 2013

A239581 Number of primitive Pythagorean triangles (x, y, z) with legs x < y < 10^n.

Original entry on oeis.org

1, 18, 179, 1788, 17861, 178600, 1786011, 17860355, 178603639, 1786036410, 17860362941

Offset: 1

Martin Renner, Mar 26 2014

A Pythagorean triangle is a right triangle with integer side lengths x, y, z forming a Pythagorean triple (x, y, z). It is called primitive, if gcd(x, y, z) = 1.

Because (x, y, z) is equivalent to (y, x, z), the total number of primitive Pythagorean triangles with legs x, y < 10^n is b(n) = 2*a(n) = 2, 36, 358, 3576, 35722, ...

			a(1) = 1, because the only primitive Pythagorean triangle with x < y < 10 is [3, 4, 5].

Eric Weisstein's World of Mathematics, Pythagorean Triangle.
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A008846, A020882, A020883, A020884, A046086, A046087, A101931, A101929, A239744, A239786.

a(6)-a(11) from Giovanni Resta, Mar 27 2014

A239744 Number of Pythagorean triangles (x, y, z) with legs x < y <= 10^n.

Original entry on oeis.org

2, 63, 1034, 14474, 185864, 2269788, 26809924, 309224756, 3503496007, 39147452729, 432599522197

Offset: 1

Martin Renner, Mar 26 2014

A Pythagorean triangle is a right triangle with integer side lengths x, y, z forming a Pythagorean triple (x, y, z).

Because (x, y, z) is equivalent to (y, x, z), the total number of Pythagorean triangles with legs x, y < 10^n is b(n) = 2*a(n) = 4, 126, 2068, 28948, 371728, ...

			a(1) = 2, because the only two Pythagorean triangles with x < y < 10 are [3, 4, 5] and [6, 8, 10].

Eric Weisstein's World of Mathematics, Pythagorean Triangle.
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A008846, A020882, A020883, A020884, A046086, A046087, A101931, A101929, A239581, A239786.

a(6)-a(11) from Giovanni Resta, Mar 27 2014

A239786 Number of Pythagorean triangles (x, y, z) with legs x < y < 10^n.

Original entry on oeis.org

2, 62, 1032, 14471, 185860, 2269783, 26809918, 309224749, 3503495999, 39147452720, 432599522187

Offset: 1

Martin Renner, Mar 26 2014

A Pythagorean triangle is a right triangle with integer side length x, y, z forming a Pythagorean triple (x, y, z).

Because (x, y, z) is equivalent to (y, x, z), the total number of Pythagorean triangles with legs x, y < 10^n is b(n) = 2*a(n) = 4, 124, 2064, 28942, ...

Eric Weisstein's World of Mathematics, Pythagorean Triangle.
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A008846, A020882, A020883, A020884, A046086, A046087, A101931, A101929, A239581, A239744.

a(5)-a(11) from Giovanni Resta, Mar 27 2014

A307880 Hypotenuses of primitive pythagorean triangles having the property that the sum and absolute difference of the shorter legs are both prime numbers.

Original entry on oeis.org

13, 17, 25, 37, 53, 65, 73, 85, 97, 109, 113, 137, 149, 193, 197, 205, 221, 233, 277, 289, 305, 317, 337, 365, 401, 425, 445, 449, 457, 485, 505, 533, 541, 613, 625, 641, 653, 673, 697, 709, 725, 757, 785, 793, 809, 821, 877, 905, 925, 949, 1009

Offset: 1

Torlach Rush, May 02 2019

nonn

			13 is a term because 12 + 5 = 17 and 12 - 5 = 7.
17 is a term because 15 + 8 = 23 and 15 - 8 = 7.
25 is a term because 24 + 7 = 31 and 24 - 7 = 17.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Subset of A008846.

Sqrt[#[[1]]^2+#[[2]]^2]&/@Select[Union[Sort/@({Times@@#,(Last[#]^2-First[ #]^2)/2}&/@(Select[Subsets[Range[1,51,2],{2}],GCD@@#==1&]))],AllTrue[ {Total[#],#[[2]]-#[[1]]},PrimeQ]&]//Union (* Harvey P. Dale, Sep 25 2022 *)

A349536 Consider a circle on the Z X Z lattice with radius equal to the Pythagorean hypotenuse h(n) (A009003); a(n) = number of Pythagorean triples inside a Pi/4 sector of the circle.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 31, 32, 33, 34, 35, 37, 38, 39, 40, 44, 45, 46, 47, 48, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 71, 75, 76, 77, 78, 79, 80, 84, 85, 86, 87, 89

Offset: 1

Alexander Kritov, Nov 21 2021

nonn

Number of Pythagorean triples with hypotenuse less than or equal to the next one.

			The count of non-primitive Pythagorean triples as they appear in order of increasing hypotenuse:
.
       Hypotenuse
   n  (A009003(n))       Sides       a(n)
  --  ------------  ---------------  ----
   1        5            (3,4)         1
   2       10            (6,8)         2
   3       13            (5,12)        3
   4       15            (9,12)        4
   5       17            (8,15)        5
   6       20           (12,16)        6
   7       25       (7,24), (15,20)    8
   8       26           (10,24)        9
   9       29           (20,21)       10

W. Sierpinski, Pythagorean Triangles, Dover Publications, 2003.

Alexander Kritov, Table of n, a(n) for n = 1..1050
Manuel Benito and Juan L. Varona, Pythagorean triangles with legs less than n, Journal of Computational and Applied Mathematics 143, (2002), pp. 117-126.
E. Eckert, The group of primitive Pythagorean triangles, Mathematics Magazine 57 (1984) 22-27.
Alexander Kritov, C code that generates b-file
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A008846, A009003.
Cf. A020882, A081804, A008846, A020883, A046086.
Cf. A349538 (extension to the full circle of Z^2 lattice).

// see enclosed main.c
for (long j=1;j< 101;++j)
{
for (long k=1;k< 101;++k)
{
if (k<=j)   // to avoid pairs (as we need 1/8 or quarter plane)
    {
          double hyp=sqrt(j*j+k*k);
          double c= (double) floor (hyp );
if   (fabs(hyp - c) < DBL_EPSILON)  arr[r++]= (long) c;
}}}
bubbleSort(arr, r);//sort by hypotenuse increase
for (long j=0;j< r;++j)
{
   if  ( arr[j] != arr[j+1] )
    {
        // write to file: j is the sequence value a[n]*2
        // arr[j] is the hypotenuse value
    }
}

Conjecture: the increment is a(n+1) - a(n) = 2^(m-1), where m is the sum of all powers of the Pythagorean primes (A002144) in the factorization of hypotenuse h(n+1) (see Eckert for PPT). However, starting from 58 the increment is 3.

A354379 Hypotenuses of Pythagorean triangles whose legs are also hypotenuse numbers (A009003).

Original entry on oeis.org

25, 50, 65, 75, 85, 89, 100, 109, 125, 130, 145, 149, 150, 169, 170, 173, 175, 178, 185, 195, 200, 205, 218, 221, 225, 229, 233, 250, 255, 260, 265, 267, 275, 289, 290, 293, 298, 300, 305, 313, 325, 327, 338, 340, 346, 349, 350, 353, 356, 365, 370, 375, 377, 390, 400

Offset: 1

Lamine Ngom, May 24 2022

nonn

If m is in sequence, so is any multiple of m. Primitive elements (terms which are not divisible by any previous term) are A354381.

			25 is in sequence since each member of the Pythagorean triple (15, 20, 25) belongs to A009003.
The Pythagorean triple (39, 80, 89) has all its terms in A009003. Hence 89 is in sequence.

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

Cf. A009003, A008846, A020882, A004613, A354381.

ishyp:= proc(n) local s; ormap(s -> s mod 4 = 1, numtheory:-factorset(n)) end proc:
filter:= proc(n) local s;
  ormap(s -> ishyp(subs(s,x)) and ishyp(subs(s,y)), [isolve(x^2+y^2=n^2)])
end proc:
select(filter, [$1..1000]); # Robert Israel, Jan 10 2023

ishyp[n_] := AnyTrue[FactorInteger[n][[All, 1]], Mod[#, 4] == 1&];
filter[n_] := AnyTrue[Solve[x^2 + y^2 == n^2, Integers], ishyp[x /. #] && ishyp[y /. #]&];
Select[Range[400], filter] (* Jean-François Alcover, May 11 2023, after Robert Israel *)

A354381 Primitive elements in A354379, being those not divisible by any previous term.

Original entry on oeis.org

25, 65, 85, 89, 109, 145, 149, 169, 173, 185, 205, 221, 229, 233, 265, 289, 293, 305, 313, 349, 353, 365, 377, 409, 421, 433, 449, 461, 481, 485, 493, 505, 509, 533, 565, 601, 613, 629, 641, 653, 677, 685, 689, 697, 709, 757, 761, 769, 773, 785, 793, 797, 821, 829, 841, 857, 877, 881, 901, 905

Offset: 1

Lamine Ngom, May 24 2022

nonn

			The primitive Pythagorean triple (39, 80, 89) has all its terms in A009003, and 89 is not divisible by any previous term. Hence 89 is in sequence.

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

Cf. A009003, A008846, A020882, A004613, A354379.

ishyp:= proc(n) local s; ormap(s -> s mod 4 = 1, numtheory:-factorset(n)) end proc:
filter:= proc(n) local s;
  ormap(s -> ishyp(subs(s,x)) and ishyp(subs(s,y)), [isolve(x^2+y^2=n^2)])
end proc:
R:= []: count:= 0:
for n from 1 while count < 100 do
  if ormap(t -> n mod t = 0, R) then next fi;
  if filter(n) then R:= [op(R),n]; count:= count+1; fi
od:
R; # Robert Israel, Jan 10 2023

ishyp[n_] := AnyTrue[ FactorInteger[n][[All, 1]], Mod[#, 4] == 1 &] ;
filter[n_] := AnyTrue[Solve[x^2 + y^2 == n^2, Integers], ishyp[x /. #] && ishyp[y /. #] &];
R = {}; count = 0;
For[n = 1, count < 100, n++, If[AllTrue[R, Mod[n, #] != 0&], If[filter[n], AppendTo[R, n]; count++]]];
R (* Jean-François Alcover, May 11 2023, after Robert Israel *)

Corrected by Robert Israel, Jan 10 2023

A068386 One-thirtieth the area of the unique Pythagorean triangle whose hypotenuse is A002144(n), the n-th prime of the form 4k+1.

Original entry on oeis.org

1, 2, 7, 7, 6, 21, 11, 44, 52, 78, 33, 91, 28, 154, 119, 187, 143, 57, 266, 91, 221, 364, 418, 136, 299, 483, 616, 323, 130, 385, 840, 897, 1020, 1155, 1071, 1235, 266, 782, 203, 986, 1638, 1190, 1653, 1683, 2046, 2387, 1463, 2002, 460, 2852, 2204, 357

Offset: 2

Lekraj Beedassy, Mar 08 2002

Every such prime p has a unique representation as p = r^2 + s^2 with 1 <= r < s. The corresponding right triangle has legs of lengths s^2 - r^2 and 2rs and area rs(s^2 - r^2). For p > 5, this is divisible by 30.
Calling A002330(n) and A002331(n) respectively u and v, we have a(n) = u*v*(u-v)*(u+v), for n > 1. - Lekraj Beedassy, Mar 12 2002
The corresponding Pythagorean triple (A, B, C) with A^2 = B^2 + C^2, (A > B > C) is given by {A002144(n), A002365(n), A002366(n)}, so that a(n) = B*C/(2*30) = A002365(n)*A002366(n)/60. - Lekraj Beedassy, Oct 27 2003

			The 7th prime of the form 4k+1 is 53 = 2^2 + 7^2. So the right triangle has sides 7^2 - 2^2 = 45, 2*2*7 = 28 and 53. Its area is 1/2 * 45 * 28 = 630, so a(7) = 630/30 = 21.

Cf. A008846, A002144.

a30[p_] := For[r=1, True, r++, If[IntegerQ[s=Sqrt[p-r^2]], Return[r s(s^2-r^2)/30]]]; a30/@Select[Prime/@Range[4, 150], Mod[ #, 4]==1&]
areat[p_]:=Module[{c=Flatten[PowersRepresentations[p,2,2]],a,b},a= First[c];b= Last[c];((b^2-a^2)(2a b))/2]; areat[#]/30&/@Select[Prime[ Range[4,200]],IntegerQ[(#-1)/4]&] (* Harvey P. Dale, Jun 21 2011 *)

Edited by Dean Hickerson, Mar 14 2002

A147718 Primes of the form prime(x)^2 + (prime(x) - 1)^2.

Original entry on oeis.org

5, 13, 41, 313, 1013, 1861, 3613, 7321, 9941, 10513, 13613, 20201, 21013, 34061, 52813, 59513, 99013, 218461, 277513, 353641, 370661, 391613, 424121, 427813, 481181, 584281, 632813, 702113, 750313, 820481, 825613, 904513, 1073113

Offset: 1

Artur Jasinski, Nov 11 2008

nonn

These primes are hypotenuses of right triangles in which all sides are natural numbers.

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

Cf. A020882, A008846.

aa = {}; Do[If[PrimeQ[Prime[x]^2 + (Prime[x] - 1)^2], AppendTo[aa, Prime[x]^2 + (Prime[x] - 1)^2]], {x, 1, 1000}]; aa

cp's OEIS Frontend

A230486 Numbers n such that n^n is representable as the sum of two nonzero squares.

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A239581 Number of primitive Pythagorean triangles (x, y, z) with legs x < y < 10^n.

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A239744 Number of Pythagorean triangles (x, y, z) with legs x < y <= 10^n.

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A239786 Number of Pythagorean triangles (x, y, z) with legs x < y < 10^n.

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A307880 Hypotenuses of primitive pythagorean triangles having the property that the sum and absolute difference of the shorter legs are both prime numbers.

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Mathematica

A349536 Consider a circle on the Z X Z lattice with radius equal to the Pythagorean hypotenuse h(n) (A009003); a(n) = number of Pythagorean triples inside a Pi/4 sector of the circle.

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A354379 Hypotenuses of Pythagorean triangles whose legs are also hypotenuse numbers (A009003).

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A354381 Primitive elements in A354379, being those not divisible by any previous term.

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