cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A380074 Short legs of Pythagorean triangles having legs that add up to a square ordered by increasing hypotenuse.

Original entry on oeis.org

21, 9, 84, 36, 133, 85, 189, 81, 189, 184, 336, 144, 400, 217, 532, 525, 340, 225, 820, 756, 324, 756, 736, 1036, 57, 1029, 564, 820, 1197, 672, 765, 441, 1344, 576, 1600, 1701, 868, 2128, 729, 2100, 1701, 1656, 1360, 1464, 2044, 900, 1513, 2541, 781, 2340, 3280
Offset: 1

Views

Author

Felix Huber, Jan 18 2025

Keywords

Comments

Corresponding hypotenuses in A380072, long legs in A380073.
Subsequence of A046083 and supersequence of A089547.

Examples

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

Links

Crossrefs

Cf. A000290, A007913, A046083, A089547, A379830, A380072, A380073.

Programs

  • Maple
    # Calculates the first 10001 terms
A380074:=proc(M)
    local i,m,p,q,r,u,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
                    u:=r*min(p^2-q^2,2*p*q);
                    L:=[op(L),seq([i^2*w,i^2*u],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;
A380074(4330);

A380072 Ordered hypotenuses of Pythagorean triangles having legs that add up to a square.

Original entry on oeis.org

35, 41, 140, 164, 205, 221, 315, 369, 389, 391, 560, 656, 689, 775, 820, 875, 884, 1025, 1189, 1260, 1476, 1556, 1564, 1565, 1625, 1715, 1739, 1781, 1845, 1855, 1989, 2009, 2240, 2624, 2756, 2835, 3100, 3280, 3321, 3500, 3501, 3519, 3536, 3865, 3869, 4100, 4105
Offset: 1

Views

Author

Felix Huber, Jan 18 2025

Keywords

Comments

Corresponding long legs in A380073, short legs in A380074.
Subsequence of A009000 and supersequence of A088319.

Examples

			35 is in the sequence because 21^2 + 28^2 = 35^2 and 21 + 28 = 7^2.
206125 is twice in the sequence because 31525^2 + 203700^2 = 206125^2 and 31525 + 203700 = 485^2 as well as 94588^2 + 183141^2 = 206125^2 and 94588 + 183141 = 527^2.

Links

Crossrefs

Cf. A000290, A007913, A009000, A088319, A379830, A380073, A380074.

Programs

  • Maple
    # Calculates the first 10001 terms
A380072:=proc(M)
    local i,m,p,q,r,w,L;
    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
                    L:=[op(L),seq(i^2*w,i=1..floor(sqrt(m/w)))]
                fi
            fi
        od
    od;
    return op(sort(L))
end proc;
A380072(4330);

A386308 Long legs of Pythagorean triples that do not have the form (u^2 - v^2, 2*u*v, u^2 + v^2) ordered by increasing hypotenuse (A386307), where u and v are positive integers.

Original entry on oeis.org

12, 20, 24, 28, 36, 40, 45, 44, 48, 52, 60, 56, 60, 72, 72, 68, 75, 63, 84, 76, 80, 90, 84, 88, 105, 92, 105, 96, 120, 120, 104, 120, 108, 112, 132, 105, 116, 120, 144, 124, 144, 135, 132, 156, 136, 150, 126, 140, 168, 168, 180, 148, 175, 165, 152, 156, 168, 180
Offset: 1

Views

Author

Felix Huber, Aug 19 2025

Keywords

Comments

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2 is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.

Examples

			The Pythagorean triple (9, 12, 15) does not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), because 15 is not a sum of two nonzero squares. Therefore 12 is a term.

Links

Crossrefs

Subsequence of A046084.
Cf. A366675, A380073, A386307, A386309, A386944.

Programs

  • Maple
    A386308:=proc(N) # To get all terms with hypotenuses <= N
    local i,l,m,u,v,r,x,y,z;
    l:={};
    m:={};
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            x:=min(2*u*v,u^2-v^2);
            y:=max(2*u*v,u^2-v^2);
            z:=u^2+v^2;
            m:=m union {[z,y,x]};
            if gcd(u,v)=1 and is(u-v,odd) then
                l:=l union {seq([i*z,i*y,i*x],i=1..N/z)}
            fi
        od
    od;
    r:=l minus m;
    return seq(r[i,2],i=1..nops(r));
end proc;
A386308(1000);

Formula

a(n) = sqrt(A386307(n)^2 - A386309(n)^2).
{A046084(n)} = {a(n)} union {A046087(n)} union {A386944(n)}.

A386944 Long legs of Pythagorean triples of the form (u^2 - v^2, 2*u*v, u^2 + v^2), ordered by increasing hypotenuse (A386943).

Original entry on oeis.org

8, 16, 24, 30, 32, 36, 48, 48, 42, 60, 70, 64, 80, 72, 96, 96, 90, 84, 108, 120, 100, 112, 126, 120, 110, 140, 135, 128, 160, 154, 168, 160, 144, 144, 192, 198, 192, 180, 182, 216, 224, 168, 216, 240, 196, 200, 234, 224, 252, 189, 240, 210, 286, 288, 220, 280, 280
Offset: 1

Views

Author

Felix Huber, Aug 24 2025

Keywords

Comments

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2 is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.

Examples

			The nonprimitive Pythagorean triple (6, 8, 10) is of the form (u^2 - v^2, 2*u*v, u^2 + v^2): From u = 3 and v = 1 follows u^2 - v^2 = 8 (long leg), 2*u*v = 6 (short leg), u^2 - v^2 = 10 (hypotenuse). Therefore, 8 is a term.

Links

Crossrefs

Subsequence of A046084.
Cf. A046087, A366675, A380073, A386308, A386943, A386945.

Programs

  • Maple
    A386944:=proc(N) # To get all hypotenuses <= N
    local i,l,u,v;
    l:=[];
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            if gcd(u,v)>1 or is(u-v,even) then
                l:=[op(l),[u^2+v^2,max(2*u*v,u^2-v^2),min(2*u*v,u^2-v^2)]]
            fi
        od
    od;
    l:=sort(l);
    return seq(l[i,2],i=1..nops(l));
end proc;
A386944(296);

Formula

a(n) = sqrt(A386943(n)^2 - A386945(n)^2).
{A046084(n)} = {a(n)} union {A046087(n)} union {A386308(n)}.

A379925 Numbers k for which nonnegative integers x and y exist such that x^2 + y^2 = k and x + y is a square.

Original entry on oeis.org

0, 1, 8, 10, 16, 41, 45, 53, 65, 81, 128, 130, 136, 146, 160, 178, 200, 226, 256, 313, 317, 325, 337, 353, 373, 397, 425, 457, 493, 533, 577, 625, 648, 650, 656, 666, 680, 698, 720, 746, 776, 810, 848, 890, 936, 986, 1040, 1098, 1160, 1201, 1205, 1213, 1225, 1226
Offset: 1

Views

Author

Felix Huber, Jan 25 2025

Keywords

Comments

Numbers k for which exists at least one solution to k = x^2 + (z^2 - x)^2 in integers x and z with x >= 0 and z >= sqrt(2*x).
Subsequence of A001481.

Examples

			10 is in the sequence because 10 = 1^2 + 3^2 and 1 + 3 = 2^2.
81 is in the sequence because 81 = 0^2 + 9^2 and 0 + 9 = 3^2.

Links

Crossrefs

Cf. A000161, A000290, A001481, A004018, A025284-A025320, A091072, A125110, A118882, A125022, A380072, A380073, A380074.

Programs

  • Maple
    # Calculates the first 10005 terms.
A379925:=proc(K)
    local i,j,L;
    L:={};
    for i from 0 to floor(sqrt((K+1)^2)/2) do
        for j from 0 to floor(sqrt((K+1)^2/2-i^2)) do
            if issqr(i+j) then
                L:=L union {i^2+j^2}
            fi
        od
    od;
    return op(L)
end proc;
A379925(1737);
  • PARI
    isok(n)=my(x=0, r=0); while(x<=sqrt(n) && r==0, if(issquare(n-x^2) && issquare(x+sqrtint(n-x^2)), r=1); x++); r; \\ Michel Marcus, Feb 10 2025

Formula

k = m^(4*j) is in the sequence for nonnegative integers m and j (not both 0) because x = 0 and z = m^j is a solution to m^(4*j) = x^2 + (z^2 - x)^2.
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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