A024352 -id:A024352 - cp's OEIS frontend

Showing 1-10 of 25 results. Next

A129861 Smallest square s such that A024352(n) + s is square.

1, 4, 9, 1, 16, 25, 4, 36, 1, 9, 64, 81, 16, 4, 121, 1, 144, 9, 36, 196, 225, 4, 16, 1, 64, 324, 25, 9, 400, 441, 100, 4, 529, 1, 576, 49, 144, 676, 9, 25, 64, 841, 4, 900, 1, 36, 16, 1089, 256, 100, 1225, 49, 1296, 25, 324, 4, 1521, 81, 144, 1681, 16, 36, 169, 81, 1936, 9, 484

Offset: 1

Andrew S. Plewe, May 23 2007

nonn

			5(5+6) = 55, smallest square = (6/2)^2 = 9
4(4+10) = 56, smallest square = (10/2)^2 = 25
3(3+16) = 57, smallest square = (16/2)^2 = 64
1(1+58) = 59, smallest square = (58/2)^2 = 841
6(6+4) = 60, smallest square = (4/2)^2 = 4
1(1+60) = 61, smallest square = (60/2)^2 = 900
7(7+2) = 63, smallest square = (2/2)^2 = 1
etc.

Cf. A024352.

y(y+e) = A024352(n), where e is the smallest even number that satisfies this equation, A(n) = (e/2)^2.

A020884 Ordered short legs of primitive Pythagorean triangles.

Original entry on oeis.org

3, 5, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 20, 21, 23, 24, 25, 27, 28, 28, 29, 31, 32, 33, 33, 35, 36, 36, 37, 39, 39, 40, 41, 43, 44, 44, 45, 47, 48, 48, 49, 51, 51, 52, 52, 53, 55, 56, 57, 57, 59, 60, 60, 60, 61, 63, 64, 65, 65, 67, 68, 68, 69, 69, 71, 72, 73, 75, 75, 76, 76, 77

Offset: 1

Clark Kimberling

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives values of A, sorted.
Union of A081874 and A081925. - Lekraj Beedassy, Jul 28 2006
Each term in this sequence is given by f(m,n) = m^2 - n^2 or g(m,n) = 2mn where m and n are relatively prime positive integers with m > n, m and n not both odd. For example, a(1) = f(2,1) = 2^2 - 1^2 = 3 and a(4) = g(4,1) = 2*4*1 = 8. - Agola Kisira Odero, Apr 29 2016
All powers of 2 greater than 4 (2^2) are terms, and are generated by the function g(m,n) = 2mn. - Torlach Rush, Nov 08 2019

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Reinhard Zumkeller)
P. Alfeld, Pythagorean Triples (broken link)
Nick Exner, Generating Pythagorean Triples. This was originally a Java applet (1998), modified by Michael McKelvey in 2001 and redone as an HTML page with JavaScript by Evan Ramos in 2014.
W. A. Kehowski, Pythagorean Triples.
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A009004, A020882, A020883, A020885, A020886. Different from A024352.
Cf. A024359 (gives the number of times n occurs).
Cf. A037213.

a020884 n = a020884_list !! (n-1)
a020884_list = f 1 1 where
   f u v | v > uu `div` 2        = f (u + 1) (u + 2)
         | gcd u v > 1 || w == 0 = f u (v + 2)
         | otherwise             = u : f u (v + 2)
         where uu = u ^ 2; w = a037213 (uu + v ^ 2)
-- Reinhard Zumkeller, Nov 09 2012

shortLegs = {}; amx = 99; Do[For[b = a + 1, b < (a^2/2), c = (a^2 + b^2)^(1/2); If[c == IntegerPart[c] && GCD[a, b, c] == 1, AppendTo[shortLegs, a]]; b = b + 2], {a, 3, amx}]; shortLegs (* Vladimir Joseph Stephan Orlovsky, Aug 07 2008 *)
Take[Union[Sort/@({Times@@#,(Last[#]^2-First[#]^2)/2}&/@(Select[Subsets[Range[1,101,2],{2}],GCD@@#==1&]))][[;;,1]],80] (* Harvey P. Dale, Feb 06 2025 *)

Extended and corrected by David W. Wilson

A181123 Numbers that are the differences of two positive cubes.

Original entry on oeis.org

0, 7, 19, 26, 37, 56, 61, 63, 91, 98, 117, 124, 127, 152, 169, 189, 208, 215, 217, 218, 271, 279, 296, 316, 331, 335, 342, 386, 387, 397, 448, 469, 485, 488, 504, 511, 513, 547, 602, 604, 631, 657, 665, 702, 721, 728, 784, 817, 819, 866, 875, 919, 936, 973

Offset: 1

T. D. Noe, Oct 06 2010

Because x^3-y^3 = (x-y)(x^2+xy+y^2), the difference of two cubes is a prime number only if x=y+1, in which case all the primes are cuban, see A002407.
The difference can be a square (see A038597), but Fermat's Last Theorem prevents the difference from ever being a cube. Beal's Conjecture implies that there are no higher odd powers in this sequence.
If n is in the sequence, it must be x^3-y^3 where 0 < y <= x < n^(1/2). - Robert Israel, Dec 24 2017

T. D. Noe and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 1000 from Noe)

Cf. A014439, A014440, A014441, A333376, A333377, A038593 (subsequence), A086120.

Cf. A024352 (squares), A147857 (4th powers), A181124-A181128 (5th to 9th powers).

N:= 10^4: # to get all terms <= N
sort(convert(select(`<=`, {0, seq(seq(x^3-y^3, y=1..x-1),x=1..floor(sqrt(N)))}, N),list)); # Robert Israel, Dec 24 2017

nn=10^5; p=3; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]
With[{nn=60},Take[Union[Abs[Flatten[Differences/@Tuples[ Range[ nn]^3,2]]]], nn]] (* Harvey P. Dale, May 11 2014 *)

list(lim)=my(v=List([0]),a3); for(a=2,sqrtint(lim\3), a3=a^3; for(b=if(a3>lim,sqrtnint(a3-lim-1,3)+1,1), a-1, listput(v,a3-b^3))); Set(v) \\ Charles R Greathouse IV, Jan 25 2018

A097268 Numbers that are both the sum of two nonzero squares and the difference of two nonzero squares.

Original entry on oeis.org

5, 8, 13, 17, 20, 25, 29, 32, 37, 40, 41, 45, 52, 53, 61, 65, 68, 72, 73, 80, 85, 89, 97, 100, 101, 104, 109, 113, 116, 117, 125, 128, 136, 137, 145, 148, 149, 153, 157, 160, 164, 169, 173, 180, 181, 185, 193, 197, 200, 205, 208, 212, 221, 225, 229, 232, 233

Offset: 1

Ray Chandler, Aug 19 2004

nonn

Intersection of A000404 (sum of squares) and A024352 (difference of squares).

Also: Numbers of the form x^2+4y^2, where x and y are positive integers. Cf. A154777, A092572, A154778 for analogous sequences. - M. F. Hasler, Jan 24 2009

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sum of Squares Function.

Cf. A000404, A024352, A097269, A097270, A097271.

isA097268(n) = forstep( b=2,sqrtint(n-1),2, issquare(n-b^2) && return(1)) \\ M. F. Hasler, Jan 24 2009

A139491 Numbers arising in A139490.

Original entry on oeis.org

3, 8, 9, 12, 15, 16, 21, 24, 40, 45, 48, 60, 72, 120, 168, 240, 840, 1848

Offset: 1

Artur Jasinski, Apr 24 2008, Apr 26 2008

nonn

M. F. Hasler, Apr 24 2008, observed that the numbers in this sequence are differences of two squares. For example: 3=2^2-1^2, 8=3^2-1^2, 9=5^2-4^2, 15=4^2-1^2, 16=5^2-3^2, 21=5^2-2^2, 24=5^2-1^2, 40=7^2-3^2, 45=7^2-2^2, 48=7^2-1^2, 60=8^2-2^2.
This sequence is a subsequence of A024352.
These numbers appear to be a subset of the idoneal numbers A000926. If so, then the sequence is probably complete. - T. D. Noe, Apr 27 2009

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A000926, A024352.

f = 200; g = 300; h = 30; j = 100; b = {}; Do[a = {}; Do[Do[If[PrimeQ[x^2 + n y^2], AppendTo[a, x^2 + n y^2]], {x, 0, g}], {y, 1, g}]; AppendTo[b, Take[Union[a], h]], {n, 1, f}]; Print[b]; c = {}; Do[a = {}; Do[Do[If[PrimeQ[n^2 + w*n*m + m^2], AppendTo[a, n^2 + w*n*m + m^2]], {n, m, g}], {m, 1, g}]; AppendTo[c, Take[Union[a], h]], {w, 1, j}]; Print[c]; bb = {}; cc = {}; Do[Do[If[b[[p]] == c[[q]], AppendTo[bb, p]; AppendTo[cc, q]], {p, 1, f}], {q, 1, j}]; Union[bb]

Extended by T. D. Noe, Apr 27 2009

A097269 Numbers that are the sum of two nonzero squares but not the difference of two nonzero squares.

Original entry on oeis.org

2, 10, 18, 26, 34, 50, 58, 74, 82, 90, 98, 106, 122, 130, 146, 162, 170, 178, 194, 202, 218, 226, 234, 242, 250, 274, 290, 298, 306, 314, 338, 346, 362, 370, 386, 394, 410, 442, 450, 458, 466, 482, 490, 514, 522, 530, 538, 554, 562, 578, 586, 610, 626, 634

Offset: 1

Ray Chandler, Aug 19 2004

nonn

Intersection of A000404 (sum of squares) and complement of A024352 (difference of squares).
Numbers of the form 4k+2 = double of an odd number, with the odd number equal to the sum of 2 squares (sequence A057653). - Jean-Christophe Hervé, Oct 24 2015
Numbers that are the sum of two odd squares. - Jean-Christophe Hervé, Oct 25 2015

			2 = 1^2 + 1^2, 10 = 1^2 + 3^2, 18 = 3^2 + 3^2.

Eric Weisstein's World of Mathematics, Sum of Squares Function.

Cf. A000404, A024352, A097268, A097270, A097271. Equals twice A057653.

Cf. A020668, A263715.

is(n)=if(n%4!=2,return(0)); my(f=factor(n/2)); for(i=1,#f[,1],if(bitand(f[i,2],1)==1&&bitand(f[i,1],3)==3, return(0))); 1 \\ Charles R Greathouse IV, May 31 2013

from itertools import count, islice
from sympy import factorint
def A097269_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n//2).items()),count(2,4))
A097269_list = list(islice(A097269_gen(),30)) # Chai Wah Wu, Jun 28 2022

A181124 Difference of two positive 5th powers.

Original entry on oeis.org

0, 31, 211, 242, 781, 992, 1023, 2101, 2882, 3093, 3124, 4651, 6752, 7533, 7744, 7775, 9031, 13682, 15783, 15961, 16564, 16775, 16806, 24992, 26281, 29643, 31744, 32525, 32736, 32767, 40951, 42242, 51273, 55924, 58025, 58806, 59017, 59048, 61051

Offset: 1

T. D. Noe, Oct 06 2010

nonn

Because x^5-y^5 = (x-y)(x^4+x^3*y+x^2*y^2+x*y^3+y^4), the difference of two 5th powers is a prime number only if x=y+1, in which case all the primes are in A121616. The number 7744 is the first of an infinite number of squares in this sequence.

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

Cf. A024352 (squares), A181123 (cubes), A147857 (4th powers), A181125-A181128 (6th to 9th powers)

nn=10^9; p=5; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]

A024359 Number of primitive Pythagorean triangles with short leg n.

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times A takes value n.
Number of times n occurs in A020884.
a(A139544(n)) = 0; a(A024352(n)) > 0. - Reinhard Zumkeller, Nov 09 2012

T. D. Noe, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020884, A024352, A024360, A024361, A132404 (where records occur), A139544.

a024359_list = f 0 1 a020884_list where
   f c u vs'@(v:vs) | u == v = f (c + 1) u vs
                    | u /= v = c : f 0 (u + 1) vs'
-- Reinhard Zumkeller, Nov 09 2012

solns[a_] := Module[{b, c, soln}, soln = Reduce[a^2 + b^2 == c^2 && a < b && c > 0 && GCD[a, b, c] == 1, {b, c}, Integers]; If[soln === False, 0, If[soln[[1, 1]] === b, 1, Length[soln]]]]; Table[solns[n], {n, 100}]
(* Second program: *)
a[n_] := Module[{s = 0, b, c, d, g}, Do[g = Quotient[n^2, d]; If[d <= g && Mod[d+g, 2] == 0, c = Quotient[d+g, 2]; b = g-c; If[n < b && GCD[b, c] == 1, s++]], {d, Divisors[n^2]}]; s]; Array[a, 100] (* Jean-François Alcover, Apr 27 2019, from PARI *)

nppt(a) = {
  my(s=0, b, c, d, g);
  fordiv(a^2, d,
    g=a^2\d;
    if(d<=g && (d+g)%2==0,
      c=(d+g)\2; b=g-c;
      if(aColin Barker, Oct 25 2015

Formula

a(n) = A024361(n) - A024360(n). - Ray Chandler, Feb 03 2020

A181128 Difference of two positive 9th powers.

Original entry on oeis.org

0, 511, 19171, 19682, 242461, 261632, 262143, 1690981, 1933442, 1952613, 1953124, 8124571, 9815552, 10058013, 10077184, 10077695, 30275911, 38400482, 40091463, 40333924, 40353095, 40353606, 93864121, 124140032, 132264603, 133955584

Offset: 1

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Author

T. D. Noe, Oct 06 2010

Keywords

nonn

Comments

No term is a prime number.

Links

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

Crossrefs

Cf. A024352 (squares), A181123 (cubes), A147857 (4th powers), A181124-A181127 (5th to 8th powers)

Programs

Mathematica

nn=10^15; p=9; Union[Reap[Do[n=i^p-j^p; If[n<=nn, Sow[n]], {i,Ceiling[(nn/p)^(1/(p-1))]}, {j,i}]][[2,1]]]

A105020 Array read by antidiagonals: row n (n >= 0) contains the numbers m^2 - n^2, m >= n+1.

Original entry on oeis.org

1, 3, 4, 5, 8, 9, 7, 12, 15, 16, 9, 16, 21, 24, 25, 11, 20, 27, 32, 35, 36, 13, 24, 33, 40, 45, 48, 49, 15, 28, 39, 48, 55, 60, 63, 64, 17, 32, 45, 56, 65, 72, 77, 80, 81, 19, 36, 51, 64, 75, 84, 91, 96, 99, 100, 21, 40, 57, 72, 85, 96, 105, 112, 117, 120, 121

Offset: 0

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Author

Andrew S. Plewe and Franklin T. Adams-Watters, Jul 11 2005

Keywords

nonn
tabl
easy

Comments

A "Goldbach Conjecture" for this sequence: when there are n terms between consecutive odd integers (2n+1) and (2n+3) for n > 0, at least one will be the product of 2 primes (not necessarily distinct). Example: n=3 for consecutive odd integers a(7)=7 and a(11)=9 and of the 3 sequence entries a(8)=12, a(9)=15 and a(10)=16 between them, one is the product of 2 primes a(9)=15=3*5. - Michael Hiebl, Jul 15 2007

A024352 gives distinct values in the array, minus the first row (1, 4, 9, 16, etc.). a(n) gives all solutions to the equation x^2 + xy = n, with y mod 2 = 0, x > 0, y >= 0. - Andrew S. Plewe, Oct 19 2007

Alternatively, triangular sequence of coefficients of Dynkin diagram weights for the Cartan groups C_n: t(n,m) = m*(2*n - m). Row sums are A002412. - Roger L. Bagula, Aug 05 2008

Examples

Array begins: 1 4 9 16 25 36 49 64 81 100 ... 3 8 15 24 35 48 63 80 99 120 ... 5 12 21 32 45 60 77 96 117 140 ... 7 16 27 40 55 72 91 112 135 160 ... 9 20 33 48 65 84 105 128 153 180 ... ... Triangle begins: 1; 3, 4; 5, 8, 9; 7, 12, 15, 16; 9, 16, 21, 24, 25; 11, 20, 27, 32, 35, 36; 13, 24, 33, 40, 45, 48, 49; 15, 28, 39, 48, 55, 60, 63, 64; 17, 32, 45, 56, 65, 72, 77, 80, 81; 19, 36, 51, 64, 75, 84, 91, 96, 99, 100;

References

R. N. Cahn, Semi-Simple Lie Algebras and Their Representations, Dover, NY, 2006, ISBN 0-486-44999-8, p. 139.

Links

G. C. Greubel, Antidiagonals n = 0..50, flattened

Crossrefs

Rows give A000290, A005563, A028347, A028560, A028566, A098603, A098847, A098848, A098849, A098850.

Columns give A005408, A008586, A016945, A008590, A017329, A008594, A008598, A008602, A008606, A000567.

Diagonals give A033428, A045944, A067725.

Cf. A000384, A002412, A024352, A105020, A128076.

Programs

Magma

[(k+1)*(2*n-k+1): k in [0..n], n in [0..15]]; // G. C. Greubel, Mar 15 2023

Mathematica

t[n_, m_]:= (n^2 - m^2); Flatten[Table[t[i, j], {i,12}, {j,i-1,0,-1}]] (* to view table *) Table[t[i, j], {j,0,6}, {i,j+1,10}]//TableForm (* Robert G. Wilson v, Jul 11 2005 *) Table[(k+1)*(2*n-k+1), {n,0,15}, {k,0,n}]//Flatten (* Roger L. Bagula, Aug 05 2008 *)

SageMath

def A105020(n,k): return (k+1)*(2*n-k+1) flatten([[A105020(n,k) for k in range(n+1)] for n in range(16)]) # G. C. Greubel, Mar 15 2023

Formula

a(n) = r^2 - (r^2 + r - m)^2/4, where r = round(sqrt(m)) and m = 2*n+2. - Wesley Ivan Hurt, Sep 04 2021

a(n) = A128076(n+1) * A105020(n+1). - Wesley Ivan Hurt, Jan 07 2022

From G. C. Greubel, Mar 15 2023: (Start)

Sum_{k=0..n} T(n, k) = A002412(n+1).

Sum_{k=0..n} (-1)^k*T(n, k) = (1/2)*((1+(-1)^n)*A000384((n+2)/2) - (1- (-1)^n)*A000384((n+1)/2)). (End)

Extensions

More terms from Robert G. Wilson v, Jul 11 2005

Showing 1-10 of 25 results. Next

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cp's OEIS Frontend

A129861 Smallest square s such that A024352(n) + s is square.

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Formula

A020884 Ordered short legs of primitive Pythagorean triangles.

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Haskell

Mathematica

Extensions

A181123 Numbers that are the differences of two positive cubes.

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Maple

Mathematica

PARI

A097268 Numbers that are both the sum of two nonzero squares and the difference of two nonzero squares.

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Keywords

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Crossrefs

Programs

PARI

A139491 Numbers arising in A139490.

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Mathematica

Extensions

A097269 Numbers that are the sum of two nonzero squares but not the difference of two nonzero squares.

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PARI

Python

A181124 Difference of two positive 5th powers.

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Mathematica

A024359 Number of primitive Pythagorean triangles with short leg n.

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Haskell

Mathematica

PARI