A046084 -id:A046084 - cp's OEIS frontend

A342491 a(n) = f(x)+f(y)+f(z), where (x,y,h) is the n-th Pythagorean triple listed in (A046083, A046084, A009000), and f(m)=A176774(m) is the smallest polygonality of m.

12, 14, 23, 12, 28, 29, 27, 20, 38, 52, 27, 22, 11, 47, 20, 49, 53, 16, 69, 81, 17, 47, 59, 59, 34, 41, 93, 32, 76, 33, 34, 121, 76, 93, 88, 33, 37, 39, 101, 102, 83, 27, 90, 52, 73, 183, 75, 37, 45, 130, 105, 15, 155, 83, 120, 54, 106, 133, 129, 15, 123, 42, 225

Offset: 1

Michel Marcus, Mar 14 2021

nonn

Inspired by (A245646, A245647, A245648), for which a(n) = 12.

Examples of lower terms: 11 for (21, 28, 35), 10 for (64, 120, 136) and 9 for (8778, 10296, 13530).

			a(1) = 12 because (3, 4, 5) are (3-, 4-, 5-) gonal numbers, and 3+4+5=12.

Michel Marcus, Table of n, a(n) for n = 1..12471 (hypotenuses <= 10000).

Cf. A009000, A046083, A046084, A176774.
Cf. A213188 (see 2nd comment).
Cf. A245646, A245647, A245648.

tp(n) = my(k=3); while( !ispolygonal(n,k), k++); k; \\ A176774
f(v) = vecsum(apply(tp, v));
list(lim) = {my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, [h, middle, small]);););); v = vecsort(Vec(v)); apply(f, v);} \\ adapted from A009000

a(n) = f(A046083(n)) + f(A046084(n)) + f(A009000(n)) where f is A176774.

A382931 Numbers k for which the Pythagorean triangle (A046083(k), A046084(k), A009000(k)) has an integer altitude.

Original entry on oeis.org

7, 19, 36, 51, 69, 88, 99, 106, 126, 147, 163, 187, 196, 208, 227, 240, 250, 273, 293, 314, 342, 361, 384, 392, 409, 434, 455, 459, 483, 504, 507, 525, 549, 552, 579, 599, 627, 649, 679, 702, 711, 718, 724, 744, 752, 775, 802, 829, 854, 879, 894, 908, 935, 960

Offset: 1

Felix Huber, Apr 11 2025

nonn

Let (a, b, c) be a primitive Pythagorean triple. Since gcd(a, b, c) = 1, all and only the Pythagorean triples (k*c*a, k*c*b, k*c^2) have an integer altitude h = (k*c*a*k*c*b)/(k*c^2) = k*a*b, where k is a positive integer.

			7 is in the sequence because the pythagorean triangle (A046083(7), A046084(7), A009000(7)) = (15, 20, 25) has the integer altitude 15*20/25 = 12.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A009000, A046083, A046084, A382932.

A382931:=proc(H) # All hypotenuses <= H.
    local a,b,c,k,p,q,L,M;
    L:=[];
    M:=[];
    for p from 2 to floor(sqrt(H-1)) do
        for q to min(p-1,floor(sqrt(H-p^2))) do
            if gcd(p,q)=1 and is(p-q,odd) then
                a:=p^2-q^2;
                b:=2*p*q;
                c:=p^2+q^2;
                for k to iquo(H,c) do
                    L:=[op(L),[k*c,k*max(a,b),k*a*b/c]]
                od
            fi
        od
    od;
    L:=sort(L);
    for k to nops(L) do
        if is(L[k,3],integer) then
           M:=[op(M),k]
        fi
    od;
    return op(M)
end proc;
A382931(1075);

A382932 a(n) is the altitude of the Pythagorean triangle (A046083(A382931(n)), A046084(A382931(n)), A009000(A382931(n))).

Original entry on oeis.org

12, 24, 36, 48, 60, 72, 60, 84, 96, 108, 120, 132, 120, 144, 156, 120, 168, 180, 192, 204, 216, 228, 240, 180, 252, 264, 276, 240, 288, 300, 168, 312, 324, 240, 336, 348, 360, 372, 384, 396, 420, 300, 408, 360, 420, 432, 444, 456, 468, 480, 360, 492, 504, 516

Offset: 1

Felix Huber, Apr 13 2025

nonn

All terms are divisible by 12. Proof: (Start)
Let (a, b, c) be a primitive Pythagorean triple. Since gcd(a, b, c) = 1, all and only the Pythagorean triples (k*c*a, k*c*b, k*c^2) have an integer altitude h = (k*c*a*k*c*b)/(k*c^2) = k*a*b, where k is a positive integer.
With a = p^2 - q^2 and b = 2*p*q follows h = 2*k*p*q*(p^2 - q^2) = k*2*p*q*(p + q)*(p - q), where p > q > 0, gcd(p,q) = 1 and p or q is even.
It is to show that p*q*(p + q)*(p - q) is divisible by 6. Since p or q is divisible by 2, it remains to show that p*q*(p + q)*(p - q) is divisible by 3.
If 3 is a divisor of p or q, p*q is divisible by 3. If p mod 3 = 1 and q mod 3 = 2 or p mod 3 = 2 and q mod 3 = 1, then p + q is divisible by 3. If p mod 3 = q mod 3 = 1 or p mod 3 = q mod 3 = 2, then p - q is divisible by 3.
It follows that all terms are divisible by 12. (End)

			a(1) = 12 because the Pythagorean triangle (A046083(A382931(1)), A046084(A382931(1)), A009000(A382931(1))) = (A046083(7), A046084(7), A009000(7)) = (15, 20, 25) has the integer altitude 15*20/25 = 12.

Felix Huber, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A008594, A009000, A046083, A046084, A382931.

A382932:=proc(H) # All hypotenuses <= H.
    local a,b,c,k,p,q,L,M;
    L:=[];
    M:=[];
    for p from 2 to floor(sqrt(H-1)) do
        for q to min(p-1,floor(sqrt(H-p^2))) do
            if gcd(p,q)=1 and is(p-q,odd) then
                a:=p^2-q^2;
                b:=2*p*q;
                c:=p^2+q^2;
                for k to iquo(H,c) do
                    L:=[op(L),[k*c,k*max(a,b),k*a*b/c]]
                od
            fi
        od
    od;
    L:=sort(L);
    for k to nops(L) do
        if is(L[k,3],integer) then
           M:=[op(M),L[k,3]]
        fi
    od;
    return op(M)
end proc;
A382932(1075);

a(n) = A046083(A382931(n))*A046084(A382931(n))/A009000(A382931(n)).

A344083 a(n) = f(x)+f(y)+f(z), where (x,y,h) is the n-th Pythagorean triple listed in (A046083, A046084, A009000), and f(m)=A176775(m) is the index of m as k-gonal number for the smallest possible k.

Original entry on oeis.org

6, 9, 7, 11, 9, 9, 12, 10, 9, 10, 9, 11, 18, 10, 16, 9, 9, 20, 9, 7, 18, 9, 18, 15, 11, 14, 7, 12, 10, 13, 12, 7, 12, 15, 12, 17, 14, 18, 13, 9, 13, 14, 15, 10, 9, 7, 9, 21, 12, 10, 15, 23, 7, 9, 12, 20, 9, 18, 17, 28, 14, 16, 7, 21, 18, 24, 21, 21, 20, 16, 25

Offset: 1

Michel Marcus, May 09 2021

nonn

6 is the minimum possible value, and A176775(3,4,5) gives this minimum.

Conjecture: there are no other Pythagorean triples that give this minimum. In other words, it is the only triple with 3 A090467 terms.

Michel Marcus, Table of n, a(n) for n = 1..12471 (hypotenuses <= 10000).

Cf. A176774, A176775, A342491.

Cf. A090466, A090467.

tp(n) = my(k=3); while( !ispolygonal(n,k), k++); k; \\ A176774
itp(n) = my(m=tp(n)); (m-4+sqrtint((m-4)^2+8*(m-2)*n)) / (2*m-4); \\ A176775
f(v) = vecsum(apply(itp, v));
list(lim) = {my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, [h, middle, small]);););); v = vecsort(Vec(v)); apply(f, v);}

A009000 Ordered hypotenuse numbers (squares are sums of 2 distinct nonzero squares).

Original entry on oeis.org

5, 10, 13, 15, 17, 20, 25, 25, 26, 29, 30, 34, 35, 37, 39, 40, 41, 45, 50, 50, 51, 52, 53, 55, 58, 60, 61, 65, 65, 65, 65, 68, 70, 73, 74, 75, 75, 78, 80, 82, 85, 85, 85, 85, 87, 89, 90, 91, 95, 97, 100, 100, 101, 102, 104, 105, 106, 109, 110, 111, 113, 115, 116, 117, 119, 120

Offset: 1

David W. Wilson

The largest member 'c' of the Pythagorean triples (a,b,c) ordered by increasing c.
If c^2 = a^2 + b^2 (a < b < c) then c^2 = (n^2 + m^2)/2 with n = b - a, m = b + a. - Zak Seidov, Mar 03 2011
Numbers n such that A083025(n) > 0, i.e., n is divisible by at least one prime of the form 4k+1. - Max Alekseyev, Oct 24 2008
A number appears only once in the sequence if and only if it is divisible by exactly one prime of the form 4k+1 with multiplicity one (cf. A084645). - Jean-Christophe Hervé, Nov 11 2013
If c^2 = a^2 + b^2 with a and b > 0, then a <> b: the sum of 2 equal squares cannot be a square because sqrt(2) is not rational. - Jean-Christophe Hervé, Nov 11 2013

W. L. Schaaf, Recreational Mathematics, A Guide To The Literature, "The Pythagorean Relationship", Chapter 6 pp. 89-99 NCTM VA 1963.
W. L. Schaaf, A Bibliography of Recreational Mathematics, Vol. 2, "The Pythagorean Relation", Chapter 6 pp. 108-113 NCTM VA 1972.
W. L. Schaaf, A Bibliography of Recreational Mathematics, Vol. 3, "Pythagorean Recreations", Chapter 6 pp. 62-6 NCTM VA 1973.

Zak Seidov and T. D. Noe, Table of n, a(n) for n = 1..10000 (Zak Seidov entered the first 1981 terms).
Anonymous, Links to Pythagorean Theorem Proofs
Henry Bottomley, Pythagoras's theorem (animated proof)
Kangourou Math Website, L'animation du theoreme de Pythagore
Ron Knott, Pythagorean Triples and Online Calculators
Ron Knott, Right-angled Triangles and Pythagoras' Theorem
I. Kobayashi et al., Pythagorean Theorem (Java Interactive Proofs, Applications and Explorations)
Mathematical Database, Poster, 7 Ways to prove the Pythagorean Theorem
B. Richmond, The Pythagorean Theorem
M. Shepperd, Web Resources on Pythagoras' Theorem
J. S. Silverman, A Friendly Introduction to Number Theory, Chapters 1 to 6 (see Chapters 2 and 3).
G. Villemin's Almanach of Numbers, Triangles & Triplets de Pythagore
Eric Weisstein's World of Mathematics, Pythagorean Triple
Index entries for sequences related to sums of squares

Cf. A009012, A009003, A024507, A004431, A046083, A046084, A004144, A083025, A084645, A084646, A084647, A084648, A084649, A006339.

A009000:=proc(N) # To get all terms <= N
    local p,q,i,L;
    L:=[];
    for p from 2 to floor(sqrt(N-1)) do
        for q to p-1 do
            if igcd(p,q)=1 and is(p-q,odd) then
                L:=[op(L),seq(i*(p^2+q^2),i=1..N/(p^2+q^2))];
            fi
        od
    od;
    return op(sort(L))
end proc:
A009000(120); # Felix Huber, Feb 10 2025

max = 120; hypotenuseQ[n_] := For[k = 1, True, k++, p = Prime[k]; Which[Mod[p, 4] == 1 && Divisible[n, p], Return[True], p > n, Return[False]]]; hypotenuses = Select[Range[max], hypotenuseQ]; red[c_] := {a, b, c} /. {ToRules[ Reduce[0 < a <= b && a^2 + b^2 == c^2, {a, b}, Integers]]}; A009000 = Flatten[red /@ hypotenuses, 1][[All, -1]] (* Jean-François Alcover, May 23 2012, after Max Alekseyev *)
Sqrt[#]&/@Flatten[Table[Total/@Select[IntegerPartitions[n^2,{2}],Length[Union[#]]==2&&AllTrue[Sqrt[#],IntegerQ]&],{n,150}]] (* Harvey P. Dale, May 25 2025 *)

list(lim)=my(v=List(),m2,s2,h2,h); for(middle=4,lim-1, m2=middle^2; for(small=1,middle, s2=small^2; if(issquare(h2=m2+s2,&h), if(h>lim, break); listput(v, h)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jun 23 2017

list(lim) = {my(lh = List()); for(u = 2, sqrtint(lim), for(v = 1, u, if (u^2+v^2 > lim, break); if ((gcd(u,v) == 1) && (0 != (u-v)%2), for (i = 1, lim, if (i*(u^2+v^2) > lim, break); /* if (u^2 - v^2 < 2*u*v, w = [i*(u^2 - v^2), i*2*u*v, i*(u^2+v^2)], w = [i*2*u*v, i*(u^2 - v^2), i*(u^2+v^2)]); */ listput(lh, i*(u^2+v^2)););););); vecsort(Vec(lh));} \\ Michel Marcus, Apr 10 2021

from math import isqrt
def aupto(limit):
  s = [i*i for i in range(1, limit+1)]
  s2 = sorted(a+b for i, a in enumerate(s) for b in s[i+1:])
  return [isqrt(k) for k in s2 if k in s]
print(aupto(120)) # Michael S. Branicky, May 10 2021

A046083 The smallest member 'a' of the Pythagorean triples (a,b,c) ordered by increasing c.

Original entry on oeis.org

3, 6, 5, 9, 8, 12, 15, 7, 10, 20, 18, 16, 21, 12, 15, 24, 9, 27, 30, 14, 24, 20, 28, 33, 40, 36, 11, 39, 33, 25, 16, 32, 42, 48, 24, 45, 21, 30, 48, 18, 51, 40, 36, 13, 60, 39, 54, 35, 57, 65, 60, 28, 20, 48, 40, 63, 56, 60, 66, 36, 15, 69, 80, 45, 56, 72, 22, 27, 75, 44, 35

Offset: 1

Eric W. Weisstein

nonn

Jean-François Alcover, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Triple.

Cf. A046084, A009000.

maxHypo = 125; hypotenuseQ[n_] := For[k = 1, True, k++, p = Prime[k]; Which[Mod[p, 4] == 1 && Divisible[n, p], Return[True], p > n, Return[False]]]; hypotenuses = Select[Range[maxHypo], hypotenuseQ]; red[c_] := {a, b, c} /. {ToRules[ Reduce[0 < a <= b && a^2 + b^2 == c^2, {a, b}, Integers]]}; Sort[Flatten[red /@ hypotenuses , 1], Last[#1] < Last[#2] &][[All, 1]] (* Jean-François Alcover, Oct 23 2012 *)

A009012 Ordered long legs of Pythagorean triangles.

Original entry on oeis.org

4, 8, 12, 12, 15, 16, 20, 21, 24, 24, 24, 28, 30, 32, 35, 36, 36, 40, 40, 42, 44, 45, 45, 48, 48, 48, 52, 55, 56, 56, 60, 60, 60, 60, 63, 63, 64, 68, 70, 72, 72, 72, 72, 75, 76, 77, 80, 80, 80, 84, 84, 84, 84, 88, 90, 90, 91, 92, 96, 96, 96, 99, 100, 104, 105, 105, 105, 105, 108

Offset: 1

David W. Wilson

nonn

Very similar to A046084!.

A009023 Long legs of Pythagorean triangles.

Original entry on oeis.org

4, 8, 12, 15, 16, 20, 21, 24, 28, 30, 32, 35, 36, 40, 42, 44, 45, 48, 52, 55, 56, 60, 63, 64, 68, 70, 72, 75, 76, 77, 80, 84, 88, 90, 91, 92, 96, 99, 100, 104, 105, 108, 110, 112, 116, 117, 120, 124, 126, 128, 132, 135, 136, 140, 143, 144, 147, 148, 150, 152, 153, 154, 156

Offset: 1

David W. Wilson

nonn

A227481(a(n)) > 1. - Reinhard Zumkeller, Oct 11 2013
This is A009012 (sorted A046084) without duplicates. - Andrey Zabolotskiy, Dec 27 2017
Does a(n)/n converge to some limit? - Benoit Cloitre, Oct 18 2009
For n = {52000, 72000, 100000}, n/a(n) = {0.499, 0.50175, 0.50428}. - Alex Ratushnyak, Jan 17 2019

Wacław Sierpiński, Pythagorean triangles, Dover books. [Benoit Cloitre, Oct 17 2009]

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A074235 (complement), A009012, A046084, A227481.

a009023 n = a009023_list !! (n-1)
a009023_list = filter ((> 1) . a227481) [1..]
-- Reinhard Zumkeller, Oct 11 2013

A380073 Long legs of Pythagorean triangles having legs that add up to a square ordered by increasing hypotenuse.

Original entry on oeis.org

28, 40, 112, 160, 156, 204, 252, 360, 340, 345, 448, 640, 561, 744, 624, 700, 816, 1000, 861, 1008, 1440, 1360, 1380, 1173, 1624, 1372, 1645, 1581, 1404, 1729, 1836, 1960, 1792, 2560, 2244, 2268, 2976, 2496, 3240, 2800, 3060, 3105, 3264, 3577, 3285, 4000, 3816

Offset: 1

Felix Huber, Jan 18 2025

nonn

Corresponding hypotenuses in A380072, short legs in A380074.

Subsequence of A046084 and supersequence of A089548.

			28 is in the sequence because 21^2 + 28^2 = 35^2 and 21 + 28 = 7^2.

Felix Huber, Table of n, a(n) for n = 1..10001
Eric Weisstein's World of Mathematics, Pythagorean Triple

Cf. A000290, A007913, A046084, A089548, A379830, A380072, A380074.

# Calculates the first 10001 terms
A380073:=proc(M)
    local i,m,p,q,r,v,w,L,F;
    L:=[];
    m:=M^2+2*M+2;
    for p from 2 to M do
        for q to p-1 do
            if gcd(p,q)=1 and (is(p,even) or is(q,even)) then
                r:=1;
                for i in ifactors(p^2-q^2+2*p*q)[2] do
                    if is(i[2],odd) then
                        r:=r*i[1]
                    fi
                od;
                w:=r*(p^2+q^2);
                if w<=m then
                    v:=r*max(p^2-q^2,2*p*q);
                    L:=[op(L),seq([i^2*w,i^2*v],i=1..floor(sqrt(m/w)))]
                fi
            fi
        od
    od;
    F:=[];
    for i in sort(L) do
        F:=[op(F),i[2]]
    od;
    return op(F)
end proc;
A380073(4330);

A057099 Middle side of a Pythagorean triangle (ordered by the product of the sides).

Original entry on oeis.org

4, 8, 12, 12, 15, 16, 24, 24, 20, 21, 24, 40, 35, 30, 28, 36, 32, 48, 60, 36, 48, 45, 40, 63, 45, 44, 84, 42, 60, 48, 72, 80, 56, 70, 60, 52, 56, 72, 112, 55, 99, 60, 77, 64, 75, 84, 96, 80, 68, 120, 63, 72, 144, 120, 96, 76, 105, 90, 72, 80, 143, 126, 120, 90, 84, 108, 91

Offset: 1

Henry Bottomley, Aug 01 2000

nonn

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

Cf. A009012, A009023, A046084, A057096.

maxShortLeg = 66; terms = 67;
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, 2]] (* Jean-François Alcover, Nov 21 2019 *)

a(n) =A057096(n)/(A057098(n)*A057100(n)) =sqrt(A057100(n)^2-A057098(n)^2)

cp's OEIS Frontend

A342491 a(n) = f(x)+f(y)+f(z), where (x,y,h) is the n-th Pythagorean triple listed in (A046083, A046084, A009000), and f(m)=A176774(m) is the smallest polygonality of m.

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A382931 Numbers k for which the Pythagorean triangle (A046083(k), A046084(k), A009000(k)) has an integer altitude.

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Maple

A382932 a(n) is the altitude of the Pythagorean triangle (A046083(A382931(n)), A046084(A382931(n)), A009000(A382931(n))).

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A344083 a(n) = f(x)+f(y)+f(z), where (x,y,h) is the n-th Pythagorean triple listed in (A046083, A046084, A009000), and f(m)=A176775(m) is the index of m as k-gonal number for the smallest possible k.

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PARI

A009000 Ordered hypotenuse numbers (squares are sums of 2 distinct nonzero squares).

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Maple

Mathematica

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Python

A046083 The smallest member 'a' of the Pythagorean triples (a,b,c) ordered by increasing c.

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Mathematica

A009012 Ordered long legs of Pythagorean triangles.

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A009023 Long legs of Pythagorean triangles.

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Haskell

A380073 Long legs of Pythagorean triangles having legs that add up to a square ordered by increasing hypotenuse.