A050797 -id:A050797 - cp's OEIS frontend

A050795 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in at least one way.

3, 9, 17, 19, 33, 35, 51, 73, 81, 99, 105, 129, 145, 147, 161, 163, 179, 195, 201, 233, 243, 273, 289, 291, 297, 339, 361, 387, 393, 451, 465, 467, 483, 489, 513, 521, 577, 579, 585, 611, 627, 649, 675, 721, 723, 739, 777, 801, 809, 819, 849, 883, 899, 915

Offset: 1

Patrick De Geest, Sep 15 1999

nonn

Analogous solutions exist for the sum of two identical squares z^2-1 = 2.r^2 (e.g. 99^2-1 = 2.70^2). Values of 'z' are the terms in sequence A001541, values of 'r' are the terms in sequence A001542.
Looking at a^2 + b^2 = c^2 - 1 modulo 4, we must have a and b even and c odd. Taking a = 2u, b = 2v and c = 2w - 1 and simplifying, we get u^2 + v^2 = w(w+1). - Franklin T. Adams-Watters, May 19 2008
If n is in this sequence, then so is n^(2^k), for all k >= 0. - Altug Alkan, Apr 13 2016

			E.g. 51^2 - 1 = 10^2 + 50^2 = 22^2 + 46^2 = 34^2 + 38^2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
E.-B. Escott, Query 2521, L'Intermédiaire des Mathématiciens, 10 (1903), 285. [Contains errors]
Index entries for sequences related to sums of squares

Cf. A050796, A050797, A001541, A001542, A001333.

Cf. A140612, A002378.

t={}; Do[i=c=1; While[iJayanta Basu, Jun 01 2013 *)
Select[Range@ 1000, Length[PowersRepresentations[#^2 - 1, 2, 2] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Apr 13 2016 *)

select( {is_A050795(n)=#qfbsolve(Qfb(1,0,1),n^2-1,2)}, [1..999]) \\ M. F. Hasler, Mar 07 2022

from itertools import islice, count
from sympy import factorint
def A050795_gen(startvalue=2): # generator of terms >= startvalue
    for k in count(max(startvalue,2)):
        if all(map(lambda d: d[0] % 4 != 3 or d[1] % 2 == 0, factorint(k**2-1).items())):
            yield k
A050795_list = list(islice(A050795_gen(),20)) # Chai Wah Wu, Mar 07 2022

a(n) = 2*A140612(n) + 1. - Franklin T. Adams-Watters, May 19 2008

{k : A025426(k^2-1)>0}. - R. J. Mathar, Mar 07 2022

A050798 Numbers n such that m = n^2 + 1 is expressible as the sum of two nonzero squares in exactly two ways.

Original entry on oeis.org

1, 7, 8, 12, 13, 17, 21, 22, 23, 27, 28, 30, 31, 33, 34, 37, 41, 42, 44, 46, 48, 50, 52, 53, 55, 58, 60, 62, 63, 64, 67, 75, 76, 77, 78, 80, 81, 86, 87, 88, 89, 91, 92, 96, 97, 100, 102, 103, 104, 105, 106, 108, 109, 111, 113, 114, 115, 119, 125, 127, 129, 135, 136

Offset: 1

Patrick De Geest, Sep 15 1999

Of course m = n^2 + 1 is the sum of two squares, by definition. Here there should be just one other way to write m as a different sum of two squares.

Let p and q be primes of the form 1+4k. Then n^2+1 must be pq or 2pq. - T. D. Noe, May 27 2008

			E.g., 111^2 + 1 = 21^2 + 109^2 only.

T. D. Noe, Table of n, a(n) for n=1..1000
Eric Weisstein, MathWorld: Sum of Squares Function
Index entries for sequences related to sums of squares

Cf. A000161, A050797, A050796.

ok[1] = True; ok[n_] := Length[ {ToRules[ Reduce[ 1 < x <= y && n^2 + 1 == x^2 + y^2, {x, y}, Integers] ] } ] == 1; Select[ Range[136], ok] (* Jean-François Alcover, Feb 16 2012 *)

Better definition from T. D. Noe, May 27 2008

A274567 Least number k such that k^2-1 is the sum of two nonzero squares in exactly n ways.

Original entry on oeis.org

3, 81, 51, 291, 1251, 339, 62499, 1971, 5201, 5001, 175781251, 7299

Offset: 1

Altug Alkan, Jun 28 2016

a(11) > 25*10^5 if it exists. - Chai Wah Wu, Jul 23 2020
From David A. Corneth, Jul 23 2020: (Start)
a(13) <= 17578125001, a(17) <= 610351562499. (End)

			a(2) = 81 because 81^2 - 1 = 28^2 + 76^2 = 44^2 + 68^2.

Cf. A000404, A005563, A016032, A025426, A050795, A050797, A054994.

a(10) from Chai Wah Wu, Jul 22 2020

A274590 Numbers n such that n^2 - 1 is the average of two nonzero squares in exactly one way.

Original entry on oeis.org

9, 19, 33, 35, 73, 99, 145, 161, 163, 195, 243, 393, 483, 513, 577, 721, 723, 1153, 1763, 2177, 2305, 2593, 4803, 5185, 5833, 6273, 6963, 7057, 7395, 8713, 9523, 9603, 10083, 12483, 13923, 14113, 15875, 17425, 17673, 19043, 19601, 20737, 26243

Offset: 1

Altug Alkan, Jun 29 2016

nonn

In other words, numbers n such that n^2 - 1 is the sum of two nonzero distinct squares in exactly one way. This explains why there are many common terms between this sequence and A050797.

			9 is a term because 9^2 - 1 = (4^2 + 12^2) / 2.

Cf. A050797.

cp's OEIS Frontend

A050795 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in at least one way.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Python

Formula

A050798 Numbers n such that m = n^2 + 1 is expressible as the sum of two nonzero squares in exactly two ways.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A274567 Least number k such that k^2-1 is the sum of two nonzero squares in exactly n ways.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A274590 Numbers n such that n^2 - 1 is the average of two nonzero squares in exactly one way.

Views

Author

Keywords

Comments

Examples

Crossrefs