A046079 -id:A046079 - cp's OEIS frontend

A277557 The ordered image of the 1-to-1 mapping of an integer ordered pair (x,y) into an integer using Cantor's pairing function, where 0 < x < y, gcd(x,y)=1 and x+y odd.

8, 18, 19, 32, 33, 34, 50, 52, 53, 72, 73, 74, 75, 76, 98, 99, 100, 101, 102, 103, 128, 131, 133, 134, 162, 163, 164, 165, 166, 167, 168, 169, 200, 201, 202, 203, 204, 205, 206, 207, 208, 242, 244, 247, 248, 250, 251, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 338

Offset: 1

Frank M Jackson, Oct 19 2016

nonn

The mapping of the ordered pair (x,y) to an integer uses Cantor's pairing function to generate the integer as (x+y)(x+y+1)/2+y. Also for every ordered pair (x,y) such that 0 < x < y, gcd(x,y)=1 and x+y odd, there exists a primitive Pythagorean triple (PPT) (a, b, c) such that a = y^2-x^2, b = 2xy, c = x^2+y^2. Therefore each term in the sequence represents a unique PPT.
Numbers n for which 0 < A025581(n) < A002262(n) and A025581(n)+A002262(n) is odd, and gcd(A025581(n), A002262(n)) = 1. [The definition expressed with A-numbers.] - Antti Karttunen, Nov 02 2016
See also the triangle T(y, x) with the values for PPTs given in A278147. - Wolfdieter Lang, Nov 24 2016

			a(5)=33 because the ordered pair (2,5) maps to 33 by Cantor's pairing function (see below) and is the 5th such occurrence. Also x=2, y=5 generates a PPT with sides (21,20,29).
Note: Cantor's pairing function is simply A001477 in its two-argument tabular form A001477(k, n) = n + (k+n)*(k+n+1)/2, thus A001477(2,5) = 5 + (2+5)*(2+5+1)/2 = 33. - _Antti Karttunen_, Nov 02 2016

Wikipedia, Cantor's pairing function, and Pythagorean triple

Cf. A001477, A002262, A025581, A243808, A277632.
Cf. A020882 (is obtained when A048147(a(n)) is sorted into ascending order), A008846 (same with duplicates removed).
Cf. A020887, A020888, A120427, A024362, A024406, A046079, A046087, A070151, A156678, A156679, A156680, A156683, A156685, A222946, A278147.

Cantor[{i_, j_}] := (i+j)(i+j+1)/2+j; getparts[n_] := Reverse@Select[Reverse[IntegerPartitions[n, {2}], 2], GCD@@#==1 &]; pairs=Flatten[Table[getparts[2n+1], {n, 1, 20}], 1]; Table[Cantor[pairs[[n]]], {n, 1, Length[pairs]}]

A056137 Number of ways in which n can be the longer leg (middle side) of an integer-sided right triangle.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 3, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 0, 2, 0, 1, 0, 1, 2, 0, 0, 3, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 4, 0, 0, 2, 1, 0, 0, 0, 1, 0, 1, 0

Offset: 1

Henry Bottomley, Jun 15 2000

nonn

Cf. A009023, A055523.

a(n) = A046079(n) - A056138(n) = A046081(n) - A046080(n) - A056138(n).

A328708 Number of non-primitive Pythagorean triples with leg n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 2, 2, 0, 2, 0, 2, 2, 1, 0, 5, 1, 1, 2, 2, 0, 4, 0, 3, 2, 1, 2, 5, 0, 1, 2, 5, 0, 4, 0, 2, 5, 1, 0, 8, 1, 2, 2, 2, 0, 3, 2, 5, 2, 1, 0, 9, 0, 1, 5, 4, 2, 4, 0, 2, 2, 4, 0, 10, 0, 1, 5, 2, 2, 4, 0, 8, 3, 1, 0, 9, 2, 1, 2, 5, 0, 7, 2, 2, 2, 1, 2, 11, 0, 2, 5, 5, 0, 4

Offset: 1

Rui Lin, Oct 26 2019

nonn

Pythagorean triple including primitive ones and non-primitive ones. For a certain n, it may be a leg in either primitive Pythagorean triple, or non-primitive Pythagorean triple, or both.

This sequence is the count of n as leg in non-primitive Pythagorean triple.

			n=3 as leg in only one primitive Pythagorean triple, (3,4,5); so a(3)=0.
n=6 as leg in only one non-primitive Pythagorean triple, (6,8,10); so a(6)=1.
n=8 as leg in one primitive Pythagorean triple (8,15,17) and in one non-primitive Pythagorean triple (6,8,10); so a(8)=1.

A. Beiler, Recreations in the Theory of Numbers. New York: Dover Publications, pp. 116-117, 1966.

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A046079, A024361.

a(n) = A046079(n) - A024361(n).

A365049 a(n) is the number of distinct parallelograms with integer sides and area n, and where at least one height is an integer.

Original entry on oeis.org

1, 1, 2, 3, 2, 4, 2, 5, 5, 4, 2, 10, 2, 4, 8, 9, 2, 9, 2, 10, 8, 4, 2, 20, 5, 4, 8, 10, 2, 16, 2, 13, 8, 4, 8, 23, 2, 4, 8, 20, 2, 16, 2, 10, 18, 4, 2, 34, 5, 9, 8, 10, 2, 16, 8, 20, 8, 4, 2, 40, 2, 4, 18, 19, 8, 16, 2, 10, 8, 16, 2, 45, 2, 4, 18, 10, 8, 16, 2, 34, 13

Offset: 1

Felix Huber, Aug 18 2023

nonn

If n is not a square, there are A000005(n)/2 rectangles with A027750(n,i)*A027750(n,j) = n, i < j. If n is a square, there are (A000005(n)-1)/2 rectangles with A027750(n,i)*A027750(n,j) = n, i < j and a square with A027750(n,(A000005(n)+1)/2)^2 = n. From these rectangles and, if present, the square, further parallelograms of equal area and integer sides can be formed. A046079(A027750(n,k)) is the number of possibilities there are for each side of the rectangle or for the side of the square.

			For area n = 9 there is one rectangle (sides of lengths: 1,9) and a square (3,3) with integer sides. From both, further parallelograms with area n = 9 and integer sides can be formed. Since (9,12,15) and (9,40,41) are the only Pythagorean triples with leg 9, from the rectangle (1,9) exactly the two further parallelograms (1,15) and (1,41) with height 9 can be formed, but no further parallelogram with height 1. Since (3,4,5) is the only Pythagorean triple with leg 3, from the square (3,3) exactly one further parallelogram (3,5) with height 3 can be formed. Therefore for area n = 9 there are a(9) = 5 distinct parallelograms with integer sides.

F. Richman, Pythagorean triples.
Wikipedia, Parallelogram.
Wikipedia, Table of divisors.

Cf. A000005, A027750, A046079, A224931, A214602.

from math import prod
from itertools import takewhile
from sympy import factorint, divisors
def A365049(n): return sum(1+(prod((e+(p&1)<<1)-1 for p, e in factorint(d).items())>>1)+(prod((e+(p&1)<<1)-1 for p, e in factorint(n//d).items())>>1 if d*dChai Wah Wu, Aug 21 2023

If n is a square, then a(n) = 1 + A046079(A027750(n, (A000005(n) + 1)/2)) + Sum_{i = 1..(A000005(n) - 1)/2} (1 + A046079(A027750(n,i)) + A046079(n/A027750(n,i)));

otherwise, a(n) = Sum_{i = 1..A000005(n)/2} (1 + A046079(A027750(n,i)) + A046079(n/A027750(n,i))).

A368041 a(n) is the least number k such that k^2 can be written as the difference of two positive squares in exactly n ways.

Original entry on oeis.org

1, 3, 8, 16, 12, 64, 128, 24, 512, 1024, 48, 4096, 72, 60, 32768, 65536, 192, 144, 524288, 384, 2097152, 4194304, 120, 16777216, 432, 1536, 134217728, 576, 3072, 1073741824, 2147483648, 240, 1152, 17179869184, 12288, 68719476736, 137438953472, 360, 1728, 1099511627776

Offset: 0

Ilya Gutkovskiy, Dec 09 2023

nonn

Index of first occurrence of n in A046079.

All the terms are of the form 2^m * A147516(k), m >= 0, k >= 1. - Amiram Eldar, Nov 08 2024

			a(2) = 8: 8^2 = 10^2 - 6^2 = 17^2 - 15^2.

Amiram Eldar, Table of n, a(n) for n = 0..999

Cf. A025487, A046079, A068314, A071571, A094191, A100073, A122842, A147516.

a(n) = min(A122842(n+1), 2*A071571(n)). - Jon E. Schoenfield, Dec 09 2023

a(26)-a(29) from Michel Marcus, Dec 09 2023

a(30)-a(39) from Jon E. Schoenfield, Dec 09 2023

A056138 Number of ways in which n can be the shorter leg (shortest side) of an integer-sided right triangle.

Original entry on oeis.org

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 3, 2, 1, 2, 1, 3, 3, 1, 1, 4, 2, 1, 3, 3, 1, 3, 1, 3, 4, 1, 3, 5, 1, 1, 4, 5, 1, 3, 1, 3, 5, 1, 1, 7, 2, 2, 4, 3, 1, 3, 3, 5, 4, 1, 1, 9, 1, 1, 5, 4, 4, 4, 1, 3, 4, 3, 1

Offset: 1

Henry Bottomley, Jun 15 2000

nonn

Cf. A009004, A055525, A055527.

a(n)=my(b);sum(c=n+2,n^2\2+1,issquare(c^2-n^2,&b) && nCharles R Greathouse IV, Jul 07 2013

a(n) = A046079(n) - A056137(n) = A046081(n) - A046080(n) - A056137(n).

A307876 a(n) is the smallest m such that there are prime(n) Pythagorean triangles with a leg (not hypotenuse) of length m, or -1 if no such m exists.

Original entry on oeis.org

8, 16, 64, 24, 4096, 60, 144, 384, 16777216, 1073741824, 240, 360, 4398046511104, 98304, 9216, 18014398509481984, 13824, 6291456, 840, 31104, 2160, 402653184, 19342813113834066795298816, 1237940039285380274899124224, 5760, 884736, 61440, 37748736, 412316860416

Offset: 1

Bob Andriesse, May 02 2019

nonn

a(n) is the smallest m such that A046079(m) = n-th prime.
All a(n) > 10^6 for 8 < n < 30 were provided by Amiram Eldar.
When prime(n) is a Sophie Germain prime (A005384), then a(n) = 2^(prime(n)+1).
a(n) = m if m is the smallest solution of the equation A046079(m) = prime(n). This equation can be solved by inversing the formula for A046079(n) given by Temple Keller.

			4096 is the smallest integer that can be the harmonic mean of two different integers in 11 different ways. A000040(5) = A046079(4096) = 11, so a(5) = 4096.

Amiram Eldar, Table of n, a(n) for n = 1..39

Cf. A000040, A046079.

Let prime(n)*2 + 1 be (2*a0 - 1)*(2*a1 + 1)*(2*a2 + 1)*(2*a3 + 1)*...*(2*ak + 1). Then a(n) = (2^a0)*(p1^a1)*...*(pk^ak).

A330657 Number of ways the n-th pentagonal number A000326(n) can be written as the difference of two positive pentagonal numbers.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 0, 2, 1, 1, 3, 1, 0, 2, 3, 0, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 2, 1, 1, 1, 1, 3, 2, 2, 2, 1, 4, 1, 0, 2, 1, 2, 1, 1, 1, 3, 2, 2, 1, 3, 1, 3, 2, 1, 6, 1, 1, 1, 3, 2

Offset: 1

Bradley Klee, Mar 01 2020

nonn

Equivalently, a(n) is the number of triples [n,k,m] with k>0 satisfying the Diophantine equation n*(3*n-1) + k*(3*k-1) - m*(3*m-1) = 0. Any such triple satisfies a triangle inequality, n+k > m. The n for which there is a triple [n,n,m] are listed in A137694. Solutions of the form [n,m-1,m] appear only when n=3*z+1, z > 0. The n for which a(n)=0 are listed in A135768.

			Isosceles case, n=5: 2*5*(3*5-1) - 7*(3*7-1) = 0.

N. J. A. Sloane et al., "sum of 2 triangular numbers is a triangular number", math-fun mailing list, Feb. 19-29, 2020.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
B. Klee, Pentathagorean triples: two easy proofs, seqfan mailing list, Mar. 20, 2020.
M. A. Nyblom, On the representation of the integers as a difference of nonconsecutive triangular numbers, Fibonacci Quarterly 39:3 (2001), pp. 256-263.

Cf. A046079, A309507, A000326.

PentaTriples[PNn_] := Sort[Select[{PNn,
      (-PNn + 3 PNn^2 + # - 3 #^2)/(6 #),
      (-PNn + 3 PNn^2 + # + 3 #^2)/(6 #)
      } & /@ Divisors[PNn*(3*PNn - 1)],
   And[And @@ (IntegerQ /@ #), And @@ (# > 0 & /@ #)] &]]
Length[PentaTriples[#]] & /@ Range[100]
a[n_] := Length@FindInstance[n > 0 && y > 0 && z > 0 &&
     n (3 n - 1) + y (3 y - 1) == z (3 z - 1), {y, z}, Integers, 10^9];
a /@ Range[100]

A353875 a(n) is the minimal n-digit number which can be the length of a side of a Pythagorean triangle in the largest number of ways.

Original entry on oeis.org

5, 60, 840, 9240, 65520, 720720, 8168160, 98017920, 931170240, 9311702400, 80313433200, 931635825120, 9626903526240, 95492672074800, 890488576177200, 9973472053184640, 87624075895836480, 876240758958364800, 9419588158802421600, 99847634483305668960

Offset: 1

Zhining Yang, Jun 26 2022

			a(2)=60 because 60 is the minimal 2-digit number which can be the length of a side of an integer-sided right triangle in 14 distinct ways, (11, 60, 61), (25, 60, 65), (32, 60, 68), (36, 48, 60), (45, 60, 75), (60, 63, 87), (60, 80, 100), (60, 91, 109), (60, 144, 156), (60, 175, 185), (60, 221, 229), (60, 297, 303), (60, 448, 452), (60, 899, 901), and 14 is the maximum number of such ways for a 2-digit number.

Cf. A046079, A046080.

from sympy import factorint
def s(n):
    f=factorint(n)
    d, q=(list(f.keys()), list(f.values()))
    (a, b, c, x)=(0, 1, 1, 0)
    if(d[0]==2):
        a, x=(0, 1)
        if q[0]>1:
             a=q[0]-1
    for p in range(x, len(d)):
        b*=(1+2*q[p])
        if d[p]%4==1:
            c*=(1+2*q[p])
    return((b-1)//2+a*b+(c-1)//2)
def a(n):
    max=0
    for i in range(1+10**(n-1), 10**n):
        if s(i)>max:
            k,max=(i,s(i))
    return(n,[k,max])
for i in range(1,6):
    print (a(i))

A318575 Areas of primitive Heron triangles with square sides.

Original entry on oeis.org

32918611718880, 284239560530875680

Offset: 1

Max Alekseyev, Aug 29 2018

			a(1) is the area of the Heron triangle with sides 1853^2, 4380^2, 4427^2.
a(2) is the area of the Heron triangle with sides 11789^2, 68104^2, 68595^2.

P. Stanica, S. Sarkar, S. S. Gupta, S. Maitra, N. Kar. Counting Heron Triangles, Integers 13 (2013), A3.
R. Rathbun, Second primitive Heron triangle with square sides found, NMBRTHRY, 2018.

Cf. A046079, A221836, A221837, A221838, A223941, A237518.

a(1) was found by Stanica et al. (2013).

a(2) was found by Randall L Rathbun (2018).

cp's OEIS Frontend

A277557 The ordered image of the 1-to-1 mapping of an integer ordered pair (x,y) into an integer using Cantor's pairing function, where 0 < x < y, gcd(x,y)=1 and x+y odd.

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A056137 Number of ways in which n can be the longer leg (middle side) of an integer-sided right triangle.

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A328708 Number of non-primitive Pythagorean triples with leg n.

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A365049 a(n) is the number of distinct parallelograms with integer sides and area n, and where at least one height is an integer.

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A368041 a(n) is the least number k such that k^2 can be written as the difference of two positive squares in exactly n ways.

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A056138 Number of ways in which n can be the shorter leg (shortest side) of an integer-sided right triangle.

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A307876 a(n) is the smallest m such that there are prime(n) Pythagorean triangles with a leg (not hypotenuse) of length m, or -1 if no such m exists.

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A330657 Number of ways the n-th pentagonal number A000326(n) can be written as the difference of two positive pentagonal numbers.

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A353875 a(n) is the minimal n-digit number which can be the length of a side of a Pythagorean triangle in the largest number of ways.

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