A050795 -id:A050795 - cp's OEIS frontend

A350978 Appears to be an erroneous version of A050795.

3, 9, 19, 31, 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

Offset: 1

N. J. A. Sloane, Mar 06 2022

dead

E.-B. Escott, Query 2521, L'Intermédiaire des Mathématiciens, 10 (1903), 285.

Thanks to Chai Wah Wu for pointing out that something was wrong, and to David Applegate and Allan C. Wechsler for identifying the correct version of this sequence.

A050799 Values of n^2 - 1 resulting from A050795.

Original entry on oeis.org

8, 80, 288, 360, 1088, 1224, 2600, 5328, 6560, 9800, 11024, 16640, 21024, 21608, 25920, 26568, 32040, 38024, 40400, 54288, 59048, 74528, 83520, 84680, 88208, 114920, 130320, 149768, 154448, 203400, 216224, 218088, 233288, 239120, 263168

Offset: 0

Patrick De Geest, Sep 15 1999

nonn

Cf. A050795.

t={}; Do[i=c=1; x=n^2-1; While[iJayanta Basu, Jun 01 2013 *)

A050796 Numbers n such that n^2 + 1 is expressible as the sum of two nonzero squares in at least one way (the trivial solution n^2 + 1 = n^2 + 1^2 is not counted).

Original entry on oeis.org

1, 7, 8, 12, 13, 17, 18, 21, 22, 23, 27, 28, 30, 31, 32, 33, 34, 37, 38, 41, 42, 43, 44, 46, 47, 48, 50, 52, 53, 55, 57, 58, 60, 62, 63, 64, 67, 68, 70, 72, 73, 75, 76, 77, 78, 80, 81, 82, 83, 86, 87, 88, 89, 91, 92, 93, 96, 97, 98, 99, 100, 102, 103, 104, 105, 106, 107

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., 41^2 + 1 = 2*29^2). Values of 'z' are the terms in sequence A002315, values of 'r' are the terms in sequence A001653.
Apart from the first term, numbers n such that (n^2)! == 0 mod (n^2 + 1)^2. - Michel Lagneau, Feb 14 2012
Numbers n such that neither n^2 + 1 nor (n^2 + 1)/2 is prime. - Charles R Greathouse IV, Feb 14 2012

			E.g., 57^2 + 1 = 15^2 + 55^2 = 21^2 + 53^2 = 35^2 + 45^2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A000129, A001653, A002315, A050795, A050798.

t={1}; Do[i=c=2; While[iJayanta Basu, Jun 01 2013 *)

is(n)=!isprime((n^2+1)/if(n%2,2,1)) \\ Charles R Greathouse IV, Feb 14 2012

A140612 Integers k such that both k and k+1 are the sum of 2 squares.

Original entry on oeis.org

0, 1, 4, 8, 9, 16, 17, 25, 36, 40, 49, 52, 64, 72, 73, 80, 81, 89, 97, 100, 116, 121, 136, 144, 145, 148, 169, 180, 193, 196, 225, 232, 233, 241, 244, 256, 260, 288, 289, 292, 305, 313, 324, 337, 360, 361, 369, 388, 400, 404, 409, 424, 441, 449, 457

Offset: 1

Franklin T. Adams-Watters, May 19 2008

Equivalently, nonnegative k such that k*(k+1) is the sum of two squares.
Also, nonnegative k such that k*(k+1)/2 is the sum of two squares. This follows easily from the "sum of two squares theorem": x is the sum of two (nonnegative) squares iff its prime factorization does not contain p^e where p == 3 (mod 4) and e is odd. - Robert Israel, Mar 26 2018
Trivially, sequence includes all positive squares.

			40 = 6^2 + 2^2, 41 = 5^2 + 4^2, so 40 is in the sequence.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A000217, A000290, A000404, A001481, A002378, A050795, A073613.

[k:k in [0..460]| forall{k+a: a in [0,1]|NormEquation(1, k+a) eq true}]; // Marius A. Burtea, Oct 08 2019

(*M6*) A1 = {}; Do[If[SquaresR[2, n (n + 1)/2] > 0, AppendTo[A1, n]], {n, 0, 1500}]; A1
Join[{0}, Flatten[Position[Accumulate[Range[500]], ?(SquaresR[2, #]> 0&)]]] (* _Harvey P. Dale, Jun 07 2015 *)
SequencePosition[Table[If[SquaresR[2,n]>0,1,0],{n,0,500}],{1,1}] [[All,1]]-1 (* Harvey P. Dale, Jul 28 2021 *)

from itertools import count, islice, starmap
from sympy import factorint
def A140612_gen(startvalue=0): # generator of terms >= startvalue
    for k in count(max(startvalue,0)):
        if all(starmap(lambda d, e: e % 2 == 0 or d % 4 != 3, factorint(k*(k+1)).items())):
            yield k
A140612_list = list(islice(A140612_gen(),20)) # Chai Wah Wu, Mar 07 2022

a(1)=0 prepended and edited by Max Alekseyev, Oct 08 2019

A050797 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in exactly one way.

Original entry on oeis.org

3, 9, 17, 19, 33, 35, 73, 145, 161, 163, 195, 243, 393, 483, 513, 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, 20737

Offset: 1

Patrick De Geest, Sep 15 1999

If the definition were changed from "nonzero squares" to "nonnegative squares", there would be just one additional term, 1. - T. D. Noe, May 27 2008

			E.g. 393^2 - 1 = 28^2 + 392^2 only.

T. D. Noe, Table of n, a(n) for n=1..300
Eric Weisstein, MathWorld: Sum of Squares Function
Index entries for sequences related to sums of squares

Cf. A050798, A050795.

Cf. A000161

twoSquaresQ[ n_] := (r = Reduce [0 < a <= b && n^2 - 1 == a^2 + b^2, {a, b}, Integers]; Head[r] === And); Select[ Range[21000], twoSquaresQ] (* Jean-François Alcover, Oct 10 2011 *)

More terms from James Sellers

A082982 Numbers k such that k, k+1 and k+2 are sums of 2 squares.

Original entry on oeis.org

0, 8, 16, 72, 80, 144, 232, 288, 360, 520, 576, 584, 800, 808, 1088, 1096, 1152, 1224, 1312, 1600, 1664, 1744, 1800, 1872, 1960, 2248, 2304, 2312, 2384, 2592, 2600, 2824, 3328, 3392, 3528, 3600, 4112, 4176, 4328, 4624, 5120, 5184, 5328, 5408, 5904, 6056

Offset: 1

Xavier Xarles (xarles(AT)mat.uab.es), May 28 2003

All terms are multiples of 8, cf. A304441. - M. F. Hasler, May 13 2018

			80 is here because 80=4^2+8^2, 81=0^2+9^2 and 82=1^2+9^2.

Ivan Neretin, Table of n, a(n) for n = 1..10000
Ajai Choudhry and Bibekananda Maji, Finite sequences of integers expressible as sums of two squares, arXiv:2310.13317 [math.NT], 2023.

Cf. A001481, A050795, A304441.

issumsq(n) = {ok = 0; for (i=0, ceil(sqrt(n/2)), if (issquare(n - i^2), return (1));); return (0);}
isok(n) = issumsq(n) && issumsq(n+1) && issumsq(n+2) \\ Michel Marcus, Jun 30 2013

is_A082982(n)={n%8==0&&is_A001481(n\8)&&is_A001481(n\2+1)&&is_A001481(n+1)} \\ using is_A001481 is much faster than the issumsq() above. - M. F. Hasler, May 13 2018

a(n) = 8*A304441(n). - M. F. Hasler, May 13 2018

More terms from Michel Marcus, Jun 30 2013

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

A159935 Least integer such that a(n)^2 - n is the sum of two nonzero squares.

Original entry on oeis.org

5, 3, 2, 4, 3, 5, 4, 3, 4, 7, 6, 4, 5, 9, 4, 5, 6, 5, 6, 6, 5, 11, 12, 5, 7, 15, 6, 8, 6, 7, 8, 6, 7, 13, 6, 8, 7, 21, 8, 7, 9, 7, 10, 12, 7, 15, 8, 7, 10, 9, 10, 8, 9, 11, 8, 9, 8, 17, 30, 8, 10, 9, 8, 9, 9, 13, 10, 18, 9, 11, 12, 9, 12, 9, 10, 10, 9, 15, 16, 9, 10, 11, 10, 10, 11, 21, 12, 10, 15

Offset: 0

Franklin T. Adams-Watters, Apr 26 2009

nonn

Cf. A008846, A050795.

issum2sq(n) = local(fm, hf); hf=0;fm=factor(n);for(i=1,matsize(fm)[1],if(fm[i,1]==2,if(fm[i,2]%2,hf=1),if(fm[i,1]%4==1,hf=1,if(fm[i, 2]%2,return(0)))));hf
minsum2sq(n) = local(k); k=1;while(!issum2sq(k^2-n),k++);k
/* Note: the issum2sq function depends on PARI returning -1 as a factor for negative n. */

A096079 Near hypotenuse numbers, i.e., n such that n^2 -+ 1 = a^2 + b^2, with a, b > 1.

Original entry on oeis.org

3, 7, 8, 9, 12, 13, 17, 18, 19, 21, 22, 23, 27, 28, 30, 31, 32, 33, 34, 35, 37, 38, 41, 42, 43, 44, 46, 47, 48, 50, 51, 52, 53, 55, 57, 58, 60, 62, 63, 64, 67, 68, 70, 72, 73, 75, 76, 77, 78, 80, 81, 82, 83, 86, 87, 88, 89, 91, 92, 93, 96, 97, 98, 99, 100, 102, 103, 104, 105

Offset: 1

Lekraj Beedassy, Jul 21 2004

nonn

When a=b, entries are those of A001333. - Lekraj Beedassy, Aug 06 2004

			8 and 9 are in the sequence because we have 8^2 + 1 = 4^2 + 7^2; 9^2 - 1 = 4^2 + 8^2.

Cf. A001481, A009003.

Union of A050795 and A050796. - Franklin T. Adams-Watters, Apr 26 2009

Extended by Ray Chandler, Jul 29 2004

cp's OEIS Frontend

A350978 Appears to be an erroneous version of A050795.

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A050799 Values of n^2 - 1 resulting from A050795.

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A050796 Numbers n such that n^2 + 1 is expressible as the sum of two nonzero squares in at least one way (the trivial solution n^2 + 1 = n^2 + 1^2 is not counted).

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A140612 Integers k such that both k and k+1 are the sum of 2 squares.

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A050797 Numbers n such that n^2 - 1 is expressible as the sum of two nonzero squares in exactly one way.

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A082982 Numbers k such that k, k+1 and k+2 are sums of 2 squares.

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PARI

PARI

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

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Extensions

A159935 Least integer such that a(n)^2 - n is the sum of two nonzero squares.

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Programs

PARI

A096079 Near hypotenuse numbers, i.e., n such that n^2 -+ 1 = a^2 + b^2, with a, b > 1.

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