Robert C. Lyons - cp's OEIS frontend

A381009 Ordered areas of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

84, 25200, 6350784, 1614708480, 412583721984, 105570270965760, 27022696873181184, 6917599389942743040, 1770891934572664848384, 453347470584212823736320, 116056897129722086198083584, 29710562123440325102508441600, 7605903676927233379495034486784, 1947111326786263531071061496954880

Offset: 1

Robert C. Lyons, Feb 12 2025

Proper subset of A024406.

Paolo Xausa, Table of n, a(n) for n = 1..400
John D. Cook, Sparse binary Pythagorean triples (2025).
H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
Wikipedia, Pythagorean triple.
Index entries for linear recurrences with constant coefficients, signature (340,-22848,348160,-1048576).

Cf. A024406.

Cf. A381005 (short legs), A381006 (long legs), A381007 (hypotenuses), A381008 (perimeters).

[(2^(4*n) + 2^(2*n+1)) * (2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1) / 2: n in [1..20]];

A381009[n_] := (3*# + 2)*(# + 2)*(# - 2)*2^(2*n - 3) & [4^n]; Array[A381009, 20] (* or *)
LinearRecurrence[{340, -22848, 348160, -1048576}, {84, 25200, 6350784, 1614708480}, 20] (* Paolo Xausa, Feb 26 2025 *)

a(n) = (2^(4*n) + 2^(2*n+1)) * (2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1) / 2

def A381009(n): return (m:=1<<(n<<1)-1)*(m-1)*(m+1)*(3*m+1)<<1 # Chai Wah Wu, Feb 13 2025

a(n) = A381005(n) * A381006(n) / 2.
a(n) = (2^(4n) + 2^(2n+1)) * (2^(4n) - 2^(4n-2) - 2^(2n) - 1) / 2.
G.f.: 12*(7 - 280*x - 24832*x^2 + 163840*x^3)/((1 - 4*x)*(1 - 16*x)*(1 - 64*x)*(1 - 256*x)). - Stefano Spezia, Feb 13 2025

A381008 Ordered perimeters of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

Original entry on oeis.org

56, 800, 12416, 197120, 3147776, 50339840, 805339136, 12885032960, 206158954496, 3298536980480, 52776566521856, 844424963686400, 13510799016329216, 216172782650654720, 3458764515968024576, 55340232229718589440, 885443715572418215936, 14167099448746374594560

Offset: 1

Robert C. Lyons, Feb 12 2025

Proper subset of A024364.

Paolo Xausa, Table of n, a(n) for n = 1..800
John D. Cook, Sparse binary Pythagorean triples (2025).
H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
Wikipedia, Pythagorean triple.
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Cf. A024364.

Cf. A381005 (short legs), A381006 (long legs), A381007 (hypotenuses), A381009 (areas).

[2^(4*n+1) + 2^(2*n+1) + 2^(4*n): n in [1..20]];

A381008[n_] := #*(3*# + 2) & [4^n]; Array[A381008, 20] (* or *)
LinearRecurrence[{20, -64}, {56, 800}, 20] (* Paolo Xausa, Feb 26 2025 *)

PARI
```
a(n) = 2^(4*n+1) + 2^(2*n+1) + 2^(4*n)
```

def A381008(n): return (m:=1<<(n<<1))*(2+3*m) # Chai Wah Wu, Feb 13 2025

a(n) = A381005(n) + A381006(n) + A381007(n).
a(n) = 2^(4n+1) + 2^(2n+1) + 2^(4n).
G.f.: 8*(7 - 40*x)/((1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025

A381007 Ordered hypotenuses of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

Original entry on oeis.org

25, 337, 5185, 82177, 1311745, 20975617, 335560705, 5368774657, 85899608065, 1374390583297, 21990236749825, 351843737665537, 5629499601321985, 90071992815845377, 1441151881832300545, 23058430096431906817, 368934881491370901505, 5902958103655775993857

Offset: 1

Robert C. Lyons, Feb 12 2025

Proper subset of A020882.
Conjecture: These Pythagorean triangles are primitive. Verified up to n=100000.
The preceding conjecture is true, since, for n>=1, the values of a,b,c are given by Euclid's formula for generating Pythagorean triples: a=2xy, b=x^2-y^2, c=x^2+y^2 with x=2^(2n) and y=2^(2n-1)+1 and x and y are coprime and x is even and y is odd. - Chai Wah Wu, Feb 13 2025

Paolo Xausa, Table of n, a(n) for n = 1..800
John D. Cook, Sparse binary Pythagorean triples (2025).
H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
Wikipedia, Pythagorean triple.
Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A020882.

Cf. A381005 (short legs), A381006 (long legs), A381008 (perimeters), A381009 (areas).

[2^(4*n) + 2^(4*n-2) + 2^(2*n) + 1: n in [1..20]];

A381007[n_] := 5*4^(2*n - 1) + 4^n + 1; Array[A381007, 20] (* or *)
LinearRecurrence[{21, -84, 64}, {25, 337, 5185}, 20] (* Paolo Xausa, Feb 26 2025 *)

a(n) = 2^(4*n) + 2^(4*n-2) + 2^(2*n) + 1

def A381007(n): return (m:=1<<(n<<1)-1)*(5*m+2)+1 # Chai Wah Wu, Feb 13 2025

a(n) = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.
a(n) = sqrt( A381005(n)^2 + A381006(n)^2 ).
G.f.: (25 - 188*x + 208*x^2)/((1 - x)*(1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025

A381006 Ordered long legs of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

Original entry on oeis.org

24, 288, 4224, 66048, 1050624, 16785408, 268468224, 4295098368, 68720001024, 1099513724928, 17592194433024, 281475010265088, 4503599761588224, 72057594574798848, 1152921506754330624, 18446744082299486208, 295147905213712564224, 4722366483007084167168

Offset: 1

Robert C. Lyons, Feb 12 2025

Proper subset of A020883.
Conjecture: These Pythagorean triangles are primitive. Verified up to n=100000.
The preceding conjecture is true, since, for n>=1, the values of a,b,c are given by Euclid's formula for generating Pythagorean triples: a=2xy, b=x^2-y^2, c=x^2+y^2 with x=2^(2n) and y=2^(2n-1)+1 and x and y are coprime and x is even and y is odd. - Chai Wah Wu, Feb 13 2025

Paolo Xausa, Table of n, a(n) for n = 1..800
John D. Cook, Sparse binary Pythagorean triples (2025).
H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
Wikipedia, Pythagorean triple.
Index entries for linear recurrences with constant coefficients, signature (20,-64).

Cf. A020883.

Cf. A381005 (short legs), A381007 (hypotenuses), A381008 (perimeters), A381009 (areas).

Magma
```
[2^(4*n) + 2^(2*n+1): n in [1..20]];
```

A381006[n_] := #*(# + 2) & [4^n]; Array[A381006, 20] (* or *)
LinearRecurrence[{20, -64}, {24, 288}, 20] (* Paolo Xausa, Feb 26 2025 *)

PARI
```
a(n) = 2^(4*n) + 2^(2*n+1)
```

def A381006(n): return (m:=1<<(n<<1))*(m+2) # Chai Wah Wu, Feb 13 2025

a(n) = 2^(4n) + 2^(2n+1).
a(n) = sqrt( A381007(n)^2 - A381005(n)^2 ).
G.f.: 24*(1 - 8*x)/((1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025

A381005 Ordered short legs of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

Original entry on oeis.org

7, 175, 3007, 48895, 785407, 12578815, 201310207, 3221159935, 51539345407, 824632672255, 13194135339007, 211106215755775, 3377699653419007, 54043195260010495, 864691127381393407, 13835058050987196415, 221360928867334750207, 3541774862083514433535, 56668397794160864657407

Offset: 1

Robert C. Lyons, Feb 12 2025

Proper subset of A020884.
Conjecture: These Pythagorean triangles are primitive. Verified up to n=100000.
The preceding conjecture is true, since, for n>=1, the values of a,b,c are given by Euclid's formula for generating Pythagorean triples: a=2xy, b=x^2-y^2, c=x^2+y^2 with x=2^(2n) and y=2^(2n-1)+1 and x and y are coprime and x is even and y is odd. - Chai Wah Wu, Feb 13 2025

Paolo Xausa, Table of n, a(n) for n = 1..800
John D. Cook, Sparse binary Pythagorean triples (2025).
H. S. Uhler, A Colossal Primitive Pythagorean Triangle, The American Mathematical Monthly, Vol. 57, No. 5 (May, 1950), pp. 331-332.
Wikipedia, Pythagorean triple.
Index entries for linear recurrences with constant coefficients, signature (21,-84,64).

Cf. A020884.

Cf. A381006 (long legs), A381007 (hypotenuses), A381008 (perimeters), A381009 (areas).

[2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1: n in [1..20]];

A381005[n_] := (3*# + 2)*(# - 2)/4 & [4^n]; Array[A381005, 20] (* or *)
LinearRecurrence[{21, -84, 64}, {7, 175, 3007}, 20] (* Paolo Xausa, Feb 26 2025 *)

a(n) = 2^(4*n) - 2^(4*n-2) - 2^(2*n) - 1

def A381005(n): return ((m:=1<<(n<<1)-1)-1)*(3*m+1) # Chai Wah Wu, Feb 13 2025

a(n) = 2^(4n) - 2^(4n-2) - 2^(2n) - 1.
a(n) = sqrt( A381007(n)^2 - A381006(n)^2 ).
G.f.: (7 + 28*x - 80*x^2)/((1 - x)*(1 - 4*x)*(1 - 16*x)). - Stefano Spezia, Feb 13 2025

A380130 For n >= 2, let b(n) = 1 if A379784(n) is 3 mod 4, 0 if A379784(n) is 1 mod 4; form the RUNS transform of {b(n), n >= 2}.

Original entry on oeis.org

1, 6, 13, 34, 87, 229, 581, 1591, 4268, 11637, 31944, 88526, 246105, 688982, 1936129, 5463517, 15470445

Offset: 1

Robert C. Lyons, Jan 12 2025

			A379784 begins 1, 5, 3, 7, 11, 19, 23, 31, 13, 17, 29, 37, ..., and the {b(n), n >= 2} sequence begins 0, 1, 1, 1, 1, 1, 1, 0, ..., whose RUNS transform is 1, 6, ...

Cf. A091237, A379783, A379784.

A379899 a(1) = 2. For n > 1, a(n) = smallest prime factor of c=a(n-1)+4 that is not in {a(1), ..., a(n-1)}; if all prime factors of c are in {a(1), ..., a(n-1)}, then we try the next value of c, which is c+4; and so on.

Original entry on oeis.org

2, 3, 7, 11, 5, 13, 17, 29, 37, 41, 53, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373

Offset: 1

Robert C. Lyons, Jan 05 2025

nonn

The following are some statistics about how many terms of the sequence are required, so that the first k primes are included:
- The first 10^2 terms include the first 95 primes.
- The first 10^3 terms include the first 697 primes.
- The first 10^4 terms include the first 6783 primes.
- The first 10^5 terms include the first 98563 primes.
Conjecture: this sequence is a permutation of the primes.

			a(2) is 3 because the prime factors of c=a(1)+4 (i.e., 6) are 2 and 3, and 2 already appears in the sequence as a(1).
a(6) is 13 because the only prime factor of c=a(5)+4 (i.e., 9) is 3 which already appears in the sequence as a(2). The next value of c (i.e., c+4) is 13, which is prime and does not already appear in the sequence.

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

Cf. A031439, A072268, A131200, A174162, A379652, A379648, A379775, A379776, A379783.

Cf. A379784, A379900, A380075, A380076.

b:= proc(n) option remember; `if`(n<1, {}, b(n-1) union {a(n)}) end:
a:= proc(n) option remember; local c, p; p:= infinity;
      for c from a(n-1)+4 by 4 while p=infinity do
        p:= min(numtheory[factorset](c) minus b(n-1)) od; p
    end: a(1):=2:
seq(a(n), n=1..200);  # Alois P. Heinz, Jan 11 2025

nn = 120; c[_] := True; j = 2; s = 4; c[2] = False;
Reap[Do[m = j + s;
  While[k = SelectFirst[FactorInteger[m][[All, 1]], c];
    ! IntegerQ[k], m += s];
c[k] = False; j = Sow[k], {nn}] ][[-1, 1]] (* Michael De Vlieger, Jan 11 2025 *)

from sympy import primefactors
seq = [2]
seq_set = set(seq)
max_seq_len=100
while len(seq) <= max_seq_len:
    c = seq[-1]
    done = False
    while not done:
        c = c + 4
        factors = primefactors(c)
        for factor in factors:
            if factor not in seq_set:
                seq.append(factor)
                seq_set.add(factor)
                done = True
                break
print(seq)

A379900 a(n) = position of prime(n) in A379899, or a(n) = -1 if prime(n) is not in A379899.

Original entry on oeis.org

1, 2, 5, 3, 4, 6, 7, 12, 13, 8, 14, 9, 10, 15, 16, 11, 17, 31, 18, 19, 32, 20, 21, 33, 34, 35, 22, 23, 36, 37, 24, 25, 38, 26, 39, 27, 40, 28, 29, 41, 30, 42, 73, 43, 44, 74, 75, 76, 77, 45, 46, 78, 47, 79, 48, 80, 49, 81, 50, 51, 82, 52, 83, 84, 53, 54, 85

Offset: 1

Robert C. Lyons, Jan 05 2025

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

Cf. A379899.

from sympy import prime, primefactors, primepi
def get_a379900(a379899_indices):
    a379900 = []
    count = 1
    while True:
        p = prime(count)
        if p not in a379899_indices:
            break
        a379900.append(a379899_indices[p])
        count += 1
    return a379900
a379899 = [2]
a379899_indices = dict()
a379899_indices[2] = 1
max_a379899_len=1000
while len(a379899) <= max_a379899_len:
    candidate = a379899[-1]
    done = False
    while not done:
        candidate = candidate + 4
        factors = primefactors(candidate)
        for factor in factors:
            if factor not in a379899_indices:
                a379899.append(factor)
                a379899_indices[factor] = len(a379899)
                done = True
                break
a379900 = get_a379900(a379899_indices)
print(a379900)

A379776 a(n) = position of prime(n) in A379775, or a(n) = -1 if prime(n) is not in A379775.

Original entry on oeis.org

1, 2, 3, 6, 5, 7, 4, 12, 25, 11, 20, 13, 40, 33, 43, 45, 27, 21, 44, 42, 14, 48, 71, 26, 8, 72, 64, 41, 112, 23, 116, 82, 111, 46, 62, 53, 49, 152, 80, 75, 37, 262, 122, 9, 99, 93, 38, 115, 52, 343, 28, 286, 22, 162, 104, 274, 36, 87, 47, 70, 171, 79, 18, 140

Offset: 1

Robert C. Lyons, Jan 02 2025

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

Cf. A379775.

from sympy import prime, primefactors, primepi
def get_a379776(a379775_indices):
    a379776 = []
    count = 1
    while True:
        p = prime(count)
        if p not in a379775_indices:
            break
        a379776.append(a379775_indices[p])
        count += 1
    return a379776
a379775 = [2]
a379775_indices = dict()
a379775_indices[2] = 1
max_a379775_len=1000
while len(a379775) <= max_a379775_len:
    candidate = a379775[-1]
    done = False
    while not done:
        candidate = 2*candidate - 1
        factors = primefactors(candidate)
        for factor in factors:
            if factor not in a379775_indices:
                a379775.append(factor)
                a379775_indices[factor] = len(a379775)
                done = True
                break
a379776 = get_a379776(a379775_indices)
print(a379776)

A379775 a(1) = 2. For n > 1, a(n) = smallest prime factor of c=2a(n-1)-1 that is not in {a(1), ..., a(n-1)}; if all prime factors of c are in {a(1), ..., a(n-1)}, then we try the next value of c, which is 2c-1; and so on.

Original entry on oeis.org

2, 3, 5, 17, 11, 7, 13, 97, 193, 769, 29, 19, 37, 73, 577, 1153, 461, 307, 613, 31, 61, 241, 113, 449, 23, 89, 59, 233, 929, 619, 1237, 2473, 43, 337, 673, 269, 179, 211, 421, 41, 107, 71, 47, 67, 53, 139, 277, 79, 157, 313, 1249, 227, 151, 601, 1201, 4801

Offset: 1

Robert C. Lyons, Jan 02 2025

nonn

The following are some statistics about how many terms of the sequence are required, so that the first k primes are included:
- The first 10^2 terms include the first 28 primes.
- The first 10^3 terms include the first 92 primes.
- The first 10^4 terms include the first 710 primes.
- The first 10^5 terms include the first 4848 primes.
- The first 10^6 terms include the first 29442 primes.
- The first 10^7 terms include the first 260324 primes.
Conjecture: this sequence is a permutation of the primes.

			a(6) is 7 because the prime factors of c=2*a(5)-1 (i.e., 21) are 3 and 7, and 3 already appears in the sequence as a(2).
a(8) is 97 because the only prime factor of c=2*a(7)-1 (i.e., 25) is 5 which already appears in the sequence as a(3). The next value of c (i.e., 2*c-1) is 49; its only prime factor is 7 which already appears in the sequence as a(6). The next value of c (i.e., 2*c-1) is 97, which is prime and does not already appear in the sequence.

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

Cf. A031439, A072268, A131200, A174162, A379652, A379648, A379776.

from sympy import primefactors
seq = [2]
seq_set = set(seq)
max_seq_len=100
while len(seq) <= max_seq_len:
    c = seq[-1]
    done = False
    while not done:
        c = 2*c-1
        factors = primefactors(c)
        for factor in factors:
            if factor not in seq_set:
                seq.append(factor)
                seq_set.add(factor)
                done = True
                break
print(seq)

cp's OEIS Frontend

User: Robert C. Lyons

A381009 Ordered areas of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

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A381008 Ordered perimeters of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

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A381007 Ordered hypotenuses of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

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A381006 Ordered long legs of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

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Magma

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A381005 Ordered short legs of the Pythagorean triangles defined by a = 2^(4n) + 2^(2n+1), b = 2^(4n) - 2^(4n-2) - 2^(2n) - 1, c = 2^(4n) + 2^(4n-2) + 2^(2n) + 1.

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Comments

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Programs

Magma

Mathematica

PARI

Python

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A380130 For n >= 2, let b(n) = 1 if A379784(n) is 3 mod 4, 0 if A379784(n) is 1 mod 4; form the RUNS transform of {b(n), n >= 2}.

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A379899 a(1) = 2. For n > 1, a(n) = smallest prime factor of c=a(n-1)+4 that is not in {a(1), ..., a(n-1)}; if all prime factors of c are in {a(1), ..., a(n-1)}, then we try the next value of c, which is c+4; and so on.

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A379775 a(1) = 2. For n > 1, a(n) = smallest prime factor of c=2a(n-1)-1 that is not in {a(1), ..., a(n-1)}; if all prime factors of c are in {a(1), ..., a(n-1)}, then we try the next value of c, which is 2c-1; and so on.