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 1-5 of 5 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

A020893 Squarefree sums of two squares; or squarefree numbers with no prime factors of the form 4k+3.

Original entry on oeis.org

1, 2, 5, 10, 13, 17, 26, 29, 34, 37, 41, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 130, 137, 145, 146, 149, 157, 170, 173, 178, 181, 185, 193, 194, 197, 202, 205, 218, 221, 226, 229, 233, 241, 257, 265, 269, 274, 277, 281, 290, 293, 298, 305, 313, 314, 317, 337, 346, 349
Offset: 1

Views

Author

Steven Finch

Keywords

Comments

Primitively but not imprimitively represented by x^2 + y^2.
The disjoint union of {1}, A003654, and A031398. - Max Alekseyev, Mar 09 2010
Squarefree members of A202057. - Artur Jasinski, Dec 10 2011
Union of A231754 and 2*A231754. Squarefree numbers whose prime factors are in A002313. - Robert Israel, Aug 23 2017
It appears that a(n) is the n-th index, k, such that f(k) = 2, where f(k) = 3*(Sum_{i=1..k} floor(i^2/k)) - k^2 (see A175908). - John W. Layman, May 16 2011

References

  • Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988; see page 123.

Links

Crossrefs

Cf. A001481, A008784, A022544, A034023, A175908.
Cf. also A000290, A010052, A005117, A084349, A002313, A231754.

Programs

  • Haskell
    a020893 n = a020893_list !! (n-1)
a020893_list = filter (\x -> any (== 1) $ map (a010052 . (x -)) $
                             takeWhile (<= x) a000290_list) a005117_list
-- Reinhard Zumkeller, May 28 2015
  • Maple
    N:= 1000: # to get all terms <= N
R:= {1,2}:
p:= 2:
do
p:= nextprime(p);
if p > N then break fi;
if p mod 4 <> 1 then next fi;
R:= R union select(`<=`,map(`*`,R,p),N);
od:
sort(convert(R,list)); # Robert Israel, Aug 23 2017
  • Mathematica
    lim = 17; t = Join[{1}, Select[Union[Flatten[Table[x^2 + y^2, {x, lim}, {y, x}]]], # < lim^2 && SquareFreeQ[#] &]]
Select[Union[Total/@Tuples[Range[0,20]^2,2]],SquareFreeQ] (* Harvey P. Dale, Jul 26 2017 *)
Block[{nn = 350, p}, p = {1, 2}~Join~Select[Prime@ Range@ PrimePi@ nn, Mod[#, 4] == 1 &]; Select[Range@ nn, And[SquareFreeQ@ #, SubsetQ[p, FactorInteger[#][[All, 1]]]] &]] (* Michael De Vlieger, Aug 23 2017 *)
(* or *)
Select[Range[350], SquareFreeQ@ # && ! MemberQ[Mod[First /@ FactorInteger@ #, 4], 3] &] (* Giovanni Resta, Aug 25 2017 *)
  • PARI
    is(n)=my(f=factor(n)); for(i=1,#f~,if(f[i,2]>1 || f[i,1]%4==3, return(0))); 1 \\ Charles R Greathouse IV, Apr 20 2015
  • Python
    from itertools import count, islice
from sympy import factorint
def A020893_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 and e == 1 for p, e in factorint(n).items()),count(1))
A020893_list = list(islice(A020893_gen(),30)) # Chai Wah Wu, Jun 28 2022

Formula

a(n) ~ k*n*sqrt(log n), where k = 2.1524249... = A013661/A064533. - Charles R Greathouse IV, Apr 20 2015

Extensions

Edited by N. J. A. Sloane, Aug 30 2017

A034024 Imprimitively but not primitively represented by x^2+y^2.

Original entry on oeis.org

4, 8, 9, 16, 18, 20, 32, 36, 40, 45, 49, 52, 64, 68, 72, 80, 81, 90, 98, 100, 104, 116, 117, 121, 128, 136, 144, 148, 153, 160, 162, 164, 180, 196, 200, 208, 212, 225, 232, 234, 242, 244, 245, 256, 260, 261, 272, 288, 292, 296, 306, 320, 324, 328
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. A001481, A008784, A022544, A034023-.

A034026 Numbers that are primitively or imprimitively represented by x^2+y^2, but not both.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 121, 122, 128, 130, 136, 137, 144, 145, 146, 148, 149, 153
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. A001481, A008784, A022544, A034023-.

A034025 Both primitively and imprimitively represented by x^2+y^2.

Original entry on oeis.org

25, 50, 125, 169, 250, 289, 325, 338, 425, 578, 625, 650, 725, 841, 845, 850, 925, 1025, 1250, 1325, 1369, 1445, 1450, 1525, 1625, 1681, 1682, 1690, 1825, 1850, 2050, 2125, 2197, 2225, 2425, 2525, 2650, 2725, 2738, 2809, 2825, 2873, 2890
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Crossrefs

Cf. A001481, A008784, A022544, A034023-.

Programs

  • Mathematica
    okQ[n_] := (xy = {x, y} /. {ToRules[Reduce[n == x^2 + y^2, {x, y}, Integers]]}; cnt = Count[xy, {x_, y_} /; GCD[x, y] == 1]; Length[xy] > cnt > 0); Reap[For[n = 1, n <= 3000, n++, If[okQ[n], Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Jan 25 2013 *)
Showing 1-5 of 5 results.

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

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