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

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

Author

Franklin T. Adams-Watters, May 19 2008

Keywords

• nonn
• easy

Comments

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.

Examples

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

Links

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

Crossrefs

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

Programs

Magma

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

Mathematica

(*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 *)

Python

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

Extensions

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