A008784 -id:A008784 - cp's OEIS frontend

A082607 a(0)=1; for n > 0, a(n) = least k not included earlier such that k*a(n-1) - 1 is a square.

1, 2, 5, 10, 17, 26, 37, 50, 65, 34, 13, 25, 41, 61, 85, 113, 145, 122, 101, 82, 293, 634, 1105, 53, 109, 185, 74, 149, 250, 377, 205, 146, 97, 58, 29, 73, 137, 221, 181, 650, 541, 442, 353, 274, 953, 2042, 3541, 5450, 409, 173, 370, 289, 218, 157, 106, 337, 698

Offset: 0

Amarnath Murthy, Apr 28 2003

Conjecture: this is a permutation of A008784. - Robert Israel, Aug 25 2025

Robert Israel, Table of n, a(n) for n = 0..10000

Contained in A008784. Cf. A082608, A082609, A082610, A082611, A082612.

N:= 10000: # for terms before the first term > N
Cands:= select(t -> numtheory:-quadres(-1,t) = 1, [$2..N]): nc:= nops(Cands):
R:= 1: r:= 1:
do
  found:= false;
  for i from 1 to nc do
    if issqr(r*Cands[i]-1) then
       found:= true;
       r:= Cands[i];
       R:= R,r;
       Cands:= subsop(i=NULL,Cands);
       nc:= nc-1;
       break
    fi
  od;
  if not found then break fi
od:
R; # Robert Israel, Aug 25 2025

l = {1}; Do[k = 1; While[MemberQ[l, k] || !IntegerQ[Sqrt[k*Last[l]-1]], k++ ]; AppendTo[l, k], {n, 50}]; l (* Ryan Propper, Jun 13 2006 *)

a=[1];print1(1",");for(n=2,100,k=1;f=1;while(f,if(issquare(k*a[n-1]-1),f=0;for(i=1,n-1,if(a[i]==k,f=1)));k++);a=concat(a,k-1);print1(k-1",")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

Corrected and extended by Ryan Propper, Jun 13 2006
Definition corrected by R. J. Mathar, Nov 12 2006
More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

A192450 Numbers k such that -1 is not a square mod k.

Original entry on oeis.org

3, 4, 6, 7, 8, 9, 11, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 27, 28, 30, 31, 32, 33, 35, 36, 38, 39, 40, 42, 43, 44, 45, 46, 47, 48, 49, 51, 52, 54, 55, 56, 57, 59, 60, 62, 63, 64, 66, 67, 68, 69, 70, 71, 72, 75, 76, 77, 78, 79, 80, 81, 83, 84, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 103, 104, 105, 107, 108

Offset: 1

José María Grau Ribas, Jul 01 2011

nonn

Contains A002145 as a subsequence.

Numbers that are divisible by 4 or by a term of A002145. - Robert Israel, May 14 2020

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A008784, A002145, A334819.

filter:= n -> n mod 4 = 0 or member(3, numtheory:-factorset(n) mod 4):
select(filter, [$1..1000]); # Robert Israel, May 14 2020

Table[If[Reduce[x^2==-1,Modulus->n]===False,n],{n,2,200}]//Union

for(n=1, 1e3, if(!issquare(Mod(-1, n)), print1(n", "))) \\ Charles R Greathouse IV, Jul 04 2011

A227781 Least number of squares which add to -1 mod n.

Original entry on oeis.org

0, 1, 2, 3, 1, 2, 2, 4, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 2, 3, 2, 2, 2, 4, 1, 1, 2, 3, 1, 2, 2, 4, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 2, 3, 2, 2, 2, 4, 2, 1, 2, 3, 1, 2, 2, 4, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 2, 3, 2, 2, 2, 4, 1, 1, 2, 3, 2, 2, 2, 4, 2, 1, 2, 3, 1, 2, 2, 4, 1, 2, 2, 3, 2, 2

Offset: 1

Charles R Greathouse IV, Jul 31 2013

nonn

Pfister proved that a(p) <= 2 for all primes p; then a(p) is called the stufe of the field Z/pZ.

Conjecture: a(n) = 4 if and only if n is divisible by 8 and a(n) = 3 if and only if n is 4 mod 8. Together with A008784 this would completely define the sequence.

			a(3) = 2: 1^2 + 1^2 = -1 mod 3.
a(15) = 2: 2^2 + 5^2 = -1 mod 15.

Albert Pfister, Zur Darstellung von -1 Als Summe von Quadraten in einem Korper, J. London Math. Society, 40 (1965), pp. 159-165.
A. R. Rajwade, Squares, Cambridge Univ. Press, 1983.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A008784, A002828.

isA008784(n)=if(n%2==0, if(n%4, n/=2, return(0))); n==1||vecmax(factor(n)[, 1]%4)==1
a(n)=if(isA008784(n),return(n>1)); if(isprime(n), return(2)); if(n%8==0, return(4)); my(N, cur, new, k=1); for(i=1,n\2,cur=N=bitor(1<<(i^2%n),N)); while(!bittest(cur,n-1), new=0; for(i=1,n\2, t=cur<<(i^2%n); t=bitor(bitand(t,(1<>n); new=bitor(new,t)); k++; cur=new); k

a(n) <= A002828(n-1) <= 4.

a(n) = 1 if and only if n > 1 is in A008784. a(4n) >= 3 for all n.

A045673 Curvatures in diagram constructed by inscribing 2 circles of curvature 0 and 1 inside circle of curvature 0, continuing indefinitely to inscribe circles wherever possible.

Original entry on oeis.org

0, 1, 4, 9, 12, 16, 24, 25, 28, 33, 36, 40, 49, 52, 57, 60, 64, 72, 73, 76, 81, 84, 88, 96, 97, 100, 105, 108, 112, 121, 124, 129, 136, 144, 145, 148, 153, 156, 160, 168, 169, 172, 177, 180, 184, 192, 193, 196, 201, 204, 216, 217, 220, 225, 228, 232

Offset: 0

Allan Wilks

nonn

Appears to be a superset of {A008784 - 1}. - Ralf Stephan, Jan 26 2005

See A189226 for additional comments, references, links, examples, and crossrefs. - Jonathan Sondow, Aug 24 2012

K. E. Stange, The sensual Apollonian circle packing, arXiv:1208.4836 [math.NT], 2012-2014; see section 14. - _Jonathan Sondow_, Aug 24 2012

Cf. A042944-A042946, A045679, A189226.

A076948 Smallest k such that nk-1 is a square, or 0 if no such number exists.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 5, 0, 0, 0, 0, 5, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 10, 0, 0, 0, 0, 5, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 10, 13, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 1, 0

Offset: 1

Amarnath Murthy, Oct 20 2002

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A008784.

Cf. A037213.

a076948 1 = 1
a076948 n = if null qs then 0 else head qs
            where qs = filter ((> 0) . a037213 . subtract 1 . (* n)) [1..n]
-- Reinhard Zumkeller, Oct 25 2015

a[n_] := Module[{r, j, k}, r = Solve[j>0 && k>0 && n k - 1 == j^2, {j, k}, Integers]; If[r === {}, Return[0], Return[k /. (r /. C[1] -> 0) // Min]]]; a[1] = 1;
Array[a, 100] (* Jean-François Alcover, Apr 27 2020 *)

a(n) = if (!issquare(Mod(-1, n)), 0, my(k=1); while (!issquare(n*k-1), k++); k); \\ Michel Marcus, Apr 27 2020

a(n) != 0 if and only if n is a term of A008784. - Joerg Arndt, Apr 27 2020

a(n) = 1 if and only if n is a term of A002522. - Bernard Schott, Apr 27 2020

Edited and extended by Robert G. Wilson v, Oct 21 2002

A090736 Number of positive integers <= n that can be expressed as a sum of 2 coprime squares > 0.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 11, 11, 11, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 17, 18, 18, 18, 18, 18, 18, 18, 18, 19

Offset: 1

Benoit Cloitre, Jan 18 2004

nonn

Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 100

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A008784, A064533, A090735.

Accumulate[Table[Boole[n > 1 && IntegerExponent[n, 2] < 2 && AllTrue[FactorInteger[n][[;; , 1]], Mod[#, 4] < 3 &]], {n, 1, 100}]] (* Amiram Eldar, May 08 2022 *)

a(n)=sum(i=1,n,if(sum(u=1,i,sum(v=1,u,if(abs(u^2+v^2-i)+abs(gcd(u,v)-1),0,1))),1,0))

a(n) is asymptotic to (3/(8*K))*n/sqrt(log(n)) where K is the Landau-Ramanujan constant (A064533).

A126949 Moduli n for which -1 is a (nontrivial) power residue for some power greater than 2, i.e., m^k == -1 (mod n) for some k > 1 and some 1 < m < n-1.

Original entry on oeis.org

5, 7, 9, 10, 11, 13, 14, 17, 18, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90

Offset: 1

Emmanuel Amiot, Mar 19 2007

The complement of A178751 (within integers > 1). - M. F. Hasler, Jun 06 2016

			19 is in the sequence because -1 == 10^9 (mod 19).

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Emmanuel Amiot, Autosimilar Melodies, J. Math. Music 2 (2008), no. 3, 157-180. DOI: 10.1080/17459730802598146.
Emmanuel Amiot, Mélodies autosimilaires (Self-Replicating Melodies) (in French).

Cf. A008784, A000010.

a126949 n = a126949_list !! (n-1)
a126949_list = filter h [1..] where
   h m = not $ null [(x, e) | x <- [2 .. m - 2], gcd x m == 1,
                              e <- [2 .. a000010 m `div` 2],
                              x ^ e `mod` m == m - 1]
-- Reinhard Zumkeller, May 23 2013

ord[x_, n_] := Module[{k = 1}, While[k <= EulerPhi[n]/2 && PowerMod[x, k, n] != n - 1, k++ ]; If[PowerMod[x, k, n] == n - 1, k, infinity]] iGeneralise[n_] := Module[{candidats = Range[n - 2]}, candidats = Select[candidats, (GCD[n, # ] == 1) &]; Select[candidats, (ord[ #, n] < n) &] ] sol = {}; Do[If[iGeneralise[n] != {}, AppendTo[sol, n]], {n, 2, 100}]

is_A126949(n)={for(x=2,n-2, gcd(x,n)>1&&next; my(t=Mod(x,n)); while(abs(centerlift(t))>1,t*=x); t==-1&&return(x))} \\ (Based on code for A178751 by Ch. Greathouse.) - M. F. Hasler, Jun 07 2016

Edited by M. F. Hasler, Jun 06 2016

A151925 Write n as a sum of positive squares a^2+b^2+c^2+... with gcd(a,b,...) = 1; a(n) = minimal number of squares needed.

Original entry on oeis.org

1, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 3, 4, 3, 3, 4, 5, 2, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 3, 4, 3, 3, 4, 5, 3, 2, 3, 4, 2, 3, 4, 5, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 3, 4, 3, 3, 4, 5, 2, 2, 3, 4, 3, 3, 4, 5, 3, 2, 3, 4, 2, 3, 4, 5, 2, 3, 3, 4, 3, 3, 4, 5, 2, 3, 3

Offset: 1

N. J. A. Sloane and Vinay Vaishampayan, Aug 06 2009, Aug 07 2009

nonn

Similar to A002828, but only now primitive representations are allowed.
Of course a(n) >= A002828(n).
From Lagrange's theorem, a(n) <= 5 (see also Estermann, Grosswald, Th. 3, p. 176).
Furthermore, it appears (and should be easy to prove) that:
a(n) = 1 iff n=1
a(n) = 2 iff n in A008784\{1}
a(n) = 3 iff n in A151926
a(n) = 4 iff n == 4 or 7 mod 8
a(n) = 5 iff n == 0 mod 8

			..... n .. a(n) ..<- Numbers when squared add to n ->
-----------------------------------------------------
......1......1......1
......2......2......1......1
......3......3......1......1......1
......4......4......1......1......1......1
......5......2......1......2
......6......3......1......1......2
......7......4......1......1......1......2
......8......5......1......1......1......1......2
......9......3......1......2......2
.....10......2......1......3
.....11......3......1......1......3
.....12......4......1......1......1......3
.....13......2......2......3
.....14......3......1......2......3
.....15......4......1......1......2......3
.....16......5......1......1......1......2......3
.....17......2......1......4
.....18......3......1......1......4
.....19......3......1......3......3
.....20......4......1......1......3......3

Estermann, T., On the representations of a number as a sum of squares, Acta Arith., 45 (1937), 93-125.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985.

N. J. A. Sloane, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Fortran program

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

R. J. Mathar, Feb 01 2017

nonn

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

Robert Israel, Table of n, a(n) for n = 1..10000

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

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

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

N. J. A. Sloane

nonn

Cf. A001481, A008784, A022544, A034023-.

cp's OEIS Frontend

A082607 a(0)=1; for n > 0, a(n) = least k not included earlier such that k*a(n-1) - 1 is a square.

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A192450 Numbers k such that -1 is not a square mod k.

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Maple

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A227781 Least number of squares which add to -1 mod n.

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A045673 Curvatures in diagram constructed by inscribing 2 circles of curvature 0 and 1 inside circle of curvature 0, continuing indefinitely to inscribe circles wherever possible.

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A076948 Smallest k such that nk-1 is a square, or 0 if no such number exists.

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Haskell

Mathematica

PARI

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Extensions

A090736 Number of positive integers <= n that can be expressed as a sum of 2 coprime squares > 0.

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Author

Keywords

References

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Programs

Mathematica

PARI

Formula

A126949 Moduli n for which -1 is a (nontrivial) power residue for some power greater than 2, i.e., m^k == -1 (mod n) for some k > 1 and some 1 < m < n-1.

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Author

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Examples

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Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A151925 Write n as a sum of positive squares a^2+b^2+c^2+... with gcd(a,b,...) = 1; a(n) = minimal number of squares needed.