login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A008784 Numbers n such that sqrt(-1) mod n exists; or, numbers n that are primitively represented by x^2 + y^2. 38
1, 2, 5, 10, 13, 17, 25, 26, 29, 34, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 125, 130, 137, 145, 146, 149, 157, 169, 170, 173, 178, 181, 185, 193, 194, 197, 202, 205, 218, 221, 226, 229, 233, 241, 250, 257, 265, 269, 274, 277, 281, 289 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Indices of nonzero entries in A076948. This follows from G. H. Hardy and E. M. Wright, see reference. - Amarnath Murthy and Vladeta Jovovic, Oct 20 2002

Numbers whose prime divisors are all congruent to 1 mod 4, with the exception of at most a single factor of 2. - Franklin T. Adams-Watters, Sep 07 2008

In appears that a(n) is the set of proper divisors of numbers of the type n^2+1. - Kaloyan Todorov (kaloyan.todorov(AT)gmail.com), Mar 25 2009. This conjecture is correct. - Franklin T. Adams-Watters, Oct 07 2009

If a(n) is a member of this sequence, then so too are all of its divisors (Euler). - Ant King, Oct 11 2010

Comment from Richard R. Forberg, Mar 21 2016: (Start)

For a given a(n) > 2, there are 2^k solutions to sqrt(-1) mod n, (for some k>=1), and 2^(k-1) solutions primitively representing a(n) by x^2 + y^2.

Record setting values for the number of solutions (i.e., the next higher k values), occur at values for a(n) given by A006278.

A224450 and A224770 give a(n) values with exactly one and exactly two solutions, respectively, primitively representing integers as x^2 + y^2.

The 2^k different solutions for sqrt(-1) mod n can written as values for j, with j<=n, such that integers r = sqrt(n*j-1). However, the set of j values (listed from smallest to largest) transform into themselves symmetrically (i.e., largest to smallest) when the solutions are written as n-r. When the same 2^k solutions are written as r-j, it is clear that only 2^(k-1) distinct and independent solutions exist. (End)

REFERENCES

B. C. Berndt & R. A. Rankin, Ramanujan: Letters and Commentary, see p. 176; AMS Providence RI 1995.

J. W. S. Cassels, Rational Quadratic Forms, Cambridge, 1978.

Dickson, Leonard Eugene; History of the Theory Of Numbers, Volume II: Diophantine Analysis, Chelsea Publishing Company, 1992, pp.230-242.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Ch. 20.2-3.

LINKS

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

P. Cho-Ho Lam, Representation of integers using a^2+b^2-dc^2,

J. Int. Seq. 18 (2015) 15.8.6, Theorems 2 and 3.

Richard J. Mathar, Construction of Bhaskara pairs, arXiv:1703.01677 [math.NT], 2017.

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

MAPLE

with(numtheory); [seq(mroot(-1, 2, p), p=1..300)];

MATHEMATICA

data=Flatten[FindInstance[x^2+y^2==# && 0<=x<=# && 0<=y<=# && GCD[x, y]==1, {x, y}, Integers]&/@Range[289], 1]; x^2+y^2/.data//Union (* Ant King, Oct 11 2010 *)

Select[Range[289], And @@ (Mod[#, 4] == 1 & ) /@ (fi = FactorInteger[#]; If[fi[[1]] == {2, 1}, Rest[fi[[All, 1]]], fi[[All, 1]]])&] (* Jean-François Alcover, Jul 02 2012, after Franklin T. Adams-Watters *)

PROG

(PARI) is(n)=if(n%2==0, if(n%4, n/=2, return(0))); n==1||vecmax(factor(n)[, 1]%4)==1 \\ Charles R Greathouse IV, May 10 2012

(PARI) list(lim)=my(v=List([1, 2]), t); lim\=1; for(x=2, sqrtint(lim-1), t=x^2; for(y=0, min(x-1, sqrtint(lim-t)), if(gcd(x, y)==1, listput(v, t+y^2)))); Set(v) \\ Charles R Greathouse IV, Sep 06 2016

(Haskell)

import Data.List.Ordered (union)

a008784 n = a008784_list !! (n-1)

a008784_list = 1 : 2 : union a004613_list (map (* 2) a004613_list)

-- Reinhard Zumkeller, Oct 25 2015

CROSSREFS

Apart from the first term, a subsequence of A000404.

Cf. A001481, A022544, A020893, A037942, A034023, A057756, A076948, A045673, A004613.

Sequence in context: A099261 A103215 A037942 * A224450 A226828 A020893

Adjacent sequences:  A008781 A008782 A008783 * A008785 A008786 A008787

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Olivier Gérard

EXTENSIONS

Checked by T. D. Noe, Apr 19 2007

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 14 11:00 EST 2018. Contains 317182 sequences. (Running on oeis4.)