cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A008784 Numbers k such that sqrt(-1) mod k exists; or, numbers that are primitively represented by x^2 + y^2.

Original entry on oeis.org

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
Offset: 1

Views

Author

N. J. A. Sloane, Olivier Gérard

Keywords

Comments

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 form m^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 term of this sequence, then so too are all of its divisors (Euler). - Ant King, Oct 11 2010
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)
Lucas uses the fact that there are no multiples of 3 in this sequence to prove that one cannot have an equilateral triangle on the points of a square lattice. - Michel Marcus, Apr 27 2020
For n > 1, terms are precisely the numbers such that there is at least one pair (m,k) where m + k = a(n), and m*k == 1 (mod a(n)), m > 0 and m <= k. - Torlach Rush, Oct 18 2020
A pair (s,t) such that s+t = a(n) and s*t == +1 (mod a(n)) as above is obtained from a square root of -1 (mod a(n)) for s and t = a(n)-s. - Joerg Arndt, Oct 24 2020
The Diophantine equation x^2 + y^2 = z^5 + z with gcd(x, y, z) = 1 has solutions iff z is a term of this sequence. See Gardiner reference, Olympiad links and A340129. - Bernard Schott, Jan 17 2021
Except for 1, numbers of the form a + b + 2*sqrt(a*b - 1) for positive integers a,b such that a*b-1 is a square. - Davide Rotondo, Nov 10 2024

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.
  • Leonard Eugene Dickson, History of the Theory Of Numbers, Volume II: Diophantine Analysis, Chelsea Publishing Company, 1992, pp.230-242.
  • A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Problem 6 pp. 63 and 167-168 (1985).
  • 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

Crossrefs

Apart from the first term, a subsequence of A000404.
Cf. A001481, A022544, A020893, A037942, A034023, A057756, A076948, A045673, A004613, A340129, A192450 (complement).

Programs

  • 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
  • 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 *)
  • 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
  • PARI
    for(n=1,300,if(issquare(Mod(-1, n)),print1(n,", "))); \\ Joerg Arndt, Apr 27 2020

Extensions

Checked by T. D. Noe, Apr 19 2007

A157228 Number of primitive inequivalent inclined square sublattices of square lattice of index n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0
Offset: 1

Views

Author

N. J. A. Sloane, Feb 25 2009

Keywords

Comments

From Andrey Zabolotskiy, May 09 2018: (Start)
Also, the number of partitions of n into 2 distinct coprime squares.
All such sublattices (including non-primitive ones) are counted in A025441.
The primitive sublattices that have the same symmetries (including the orientation of the mirrors) as the parent lattice are not counted here; they are counted in A019590, and all such sublattices (including non-primitive ones) are counted in A053866.
For n > 2, equals A193138. (End)

Links

Crossrefs

Cf. A193138, A145393 (all sublattices of the square lattice), A025441, A019590, A053866, A157226, A157230, A157231, A000089, A304182, A224450, A224770, A281877, A024362.

Formula

a(n) = (A000089(n) - A019590(n)) / 2. - Andrey Zabolotskiy, May 09 2018
a(n) = 1 if n>2 is in A224450, a(n) = 2 if n is in A224770, a(n) is a higher power of 2 if n is in A281877 (first time reaches 8 at n = 32045). - Andrey Zabolotskiy, Sep 30 2018
a(n) = b(n) for odd n, a(n) = b(n/2) for even n, where b(n) = A024362(n). - Andrey Zabolotskiy, Jan 23 2022

Extensions

New name and more terms from Andrey Zabolotskiy, May 09 2018

A224770 Numbers that are the primitive sum of two squares in exactly two ways.

Original entry on oeis.org

65, 85, 130, 145, 170, 185, 205, 221, 265, 290, 305, 325, 365, 370, 377, 410, 425, 442, 445, 481, 485, 493, 505, 530, 533, 545, 565, 610, 629, 650, 685, 689, 697, 725, 730, 745, 754, 785, 793, 845, 850, 865, 890, 901, 905, 925, 949, 962, 965, 970
Offset: 1

Views

Author

Wolfdieter Lang, Apr 18 2013

Keywords

Comments

These are the increasingly ordered numbers a(n) which satisfy A193138(a(n)) = 2.
Neither the order of the squares nor the signs of the numbers to be squared are taken into account. The two squares are necessarily distinct and each is nonzero.
This sequence is a proper subsequence of A000404.

Examples

			n=1,   65:  (1, 8),  (4, 7),
n=2,   85:  (2, 9),  (6, 7),
n=3,  130:  (3, 11), (7, 9),
n=4,  145:  (1, 12), (8, 9),
n=5,  170:  (1, 13), (7, 11),
n=6,  185:  (4, 13), (8, 11),
n=7,  205:  (3, 14), (6, 13),
n=8,  221:  (5, 14), (10, 11),
n=9,  265:  (3, 16), (11, 12),
n=10, 290:  (1, 17), (11, 13).

Links

Crossrefs

Cf. A224450 (one way), A193138 (multiplicities), A000404, A024509.

Programs

  • Mathematica
    nn = 35; t = Sort[Select[Flatten[Table[If[GCD[a, b] == 1, a^2 + b^2, 0], {a, nn}, {b, a, nn}]], 0 < # <= nn^2 &]]; Transpose[Select[Tally[t], #[[2]] == 2 &]][[1]] (* T. D. Noe, Apr 20 2013 *)

Formula

a(n) = a^2 + b^2, a and integers, 0 < a < b and gcd(a,b) = 1 in exactly two ways. These representations of a(n) are denoted by two different pairs (a,b).

A281877 Numbers that are a primitive sum of two squares in more than 2 ways.

Original entry on oeis.org

1105, 1885, 2210, 2405, 2465, 2665, 3145, 3445, 3485, 3770, 3965, 4505, 4745, 4810, 4930, 5185, 5330, 5365, 5525, 5785, 5945, 6205, 6290, 6305, 6409, 6565, 6890, 6970, 7085, 7345, 7565, 7585, 7685, 7930, 8177, 8245, 8585, 8845, 8905, 9010, 9061, 9265, 9425, 9490, 9605, 9685, 9805
Offset: 1

Views

Author

R. J. Mathar, Feb 01 2017

Keywords

Comments

"Primitive" means that x and y are coprime in the representations x^2+y^2.

Links

Crossrefs

Cf. A224450 (exactly 1 way), A224770 (exactly 2 ways), A008784, A097102.

Programs

  • Maple
    N:= 10^4: # to get all terms <= N
V:= Vector(N):
for a from 1 to floor(sqrt(N)) do
  for b from 1 to min(a, floor(sqrt(N-a^2))) do
    if igcd(a,b) > 1 then next fi;
    r:= a^2 + b^2;
    V[r]:= V[r]+1;
od od:
select(n -> V[n] > 2, [$1..N]); # Robert Israel, Feb 07 2017
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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