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.
%I A000224 #105 Apr 27 2025 01:05:12
%S A000224 1,2,2,2,3,4,4,3,4,6,6,4,7,8,6,4,9,8,10,6,8,12,12,6,11,14,11,8,15,12,
%T A000224 16,7,12,18,12,8,19,20,14,9,21,16,22,12,12,24,24,8,22,22,18,14,27,22,
%U A000224 18,12,20,30,30,12,31,32,16,12,21,24,34,18,24,24,36,12
%N A000224 Number of squares mod n.
%C A000224 For any n > 2, there are quadratic nonresidues mod n, so a(n) < n. - _Charles R Greathouse IV_, Oct 28 2022
%C A000224 Conjecture: n^2 == 1 (mod a(n)*(a(n)-1)) if and only if n is an odd prime. - _Thomas Ordowski_, Apr 13 2025
%C A000224 This conjecture holds at least up to n = 10^8. - _Michel Marcus_, Apr 13 2025
%H A000224 T. D. Noe, <a href="/A000224/b000224.txt">Table of n, a(n) for n = 1..10000</a>
%H A000224 Imanuel Chen and Michael Z. Spivey, <a href="http://soundideas.pugetsound.edu/summer_research/238">Integral Generalized Binomial Coefficients of Multiplicative Functions</a>, Preprint 2015; Summer Research Paper 238, Univ. Puget.
%H A000224 Steven R. Finch and Pascal Sebah, <a href="https://arxiv.org/abs/math/0604465">Squares and Cubes Modulo n</a>, arXiv:math/0604465 [math.NT], 2006-2016.
%H A000224 Shuguang Li, <a href="http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-article-aav86i2p113bwm">On the number of elements with maximal order in the multiplicative group modulo n</a>, Acta Arithm. 86 (2) (1998) 113, see proof of theorem 2.1.
%H A000224 Param Parekh, Paavan Parekh, Sourav Deb, and Manish K. Gupta, <a href="https://arxiv.org/abs/2310.11768">On the Classification of Weierstrass Elliptic Curves over Z_n</a>, arXiv:2310.11768 [cs.CR], 2023. See p. 6.
%H A000224 E. J. F. Primrose, <a href="http://dx.doi.org/10.2307/3617445">The number of quadratic residues mod m</a>, Math. Gaz. v. 61 (1977) n. 415, 60-61.
%H A000224 Walter D. Stangl, <a href="http://www.jstor.org/stable/2690536">Counting Squares in Z_n</a>, Math. Mag. 69 (1996) 285-289.
%F A000224 a(n) = A105612(n) + 1.
%F A000224 Multiplicative with a(p^e) = floor(p^e/6) + 2 if p = 2; floor(p^(e+1)/(2p + 2)) + 1 if p > 2. - _David W. Wilson_, Aug 01 2001
%F A000224 a(2^n) = A023105(n). a(3^n) = A039300(n). a(5^n) = A039302(n). a(7^n) = A039304(n). - _R. J. Mathar_, Sep 28 2017
%F A000224 Sum_{k=1..n} a(k) ~ c * n^2/sqrt(log(n)), where c = (17/(32*sqrt(Pi))) * Product_{p prime} (1 - (p^2+2)/(2*(p^2+1)*(p+1))) * (1-1/p)^(-1/2) = 0.37672933209687137604... (Finch and Sebah, 2006). - _Amiram Eldar_, Oct 18 2022
%F A000224 If p is an odd prime, then a(p) = (p + 1)/2. - _Thomas Ordowski_, Apr 09 2025
%e A000224 The sequence of squares (A000290) modulo 10 reads 0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, 1, 4, 9, 6, 5, 6, 9, 4, 1,... and this reduced sequence contains a(10) = 6 different values, {0,1,4,5,6,9}. - _R. J. Mathar_, Oct 10 2014
%p A000224 A000224 := proc(m)
%p A000224 {seq( modp(b^2,m),b=0..m-1) };
%p A000224 nops(%) ;
%p A000224 end proc: # _Emeric Deutsch_
%p A000224 # 2nd implementation
%p A000224 A000224 := proc(n)
%p A000224 local a,ifs,f,p,e,c ;
%p A000224 a := 1 ;
%p A000224 ifs := ifactors(n)[2] ;
%p A000224 for f in ifs do
%p A000224 p := op(1,f) ;
%p A000224 e := op(2,f) ;
%p A000224 if p = 2 then
%p A000224 if type(e,'odd') then
%p A000224 a := a*(2^(e-1)+5)/3 ;
%p A000224 else
%p A000224 a := a*(2^(e-1)+4)/3 ;
%p A000224 end if;
%p A000224 else
%p A000224 if type(e,'odd') then
%p A000224 c := 2*p+1 ;
%p A000224 else
%p A000224 c := p+2 ;
%p A000224 end if;
%p A000224 a := a*(p^(e+1)+c)/2/(p+1) ;
%p A000224 end if;
%p A000224 end do:
%p A000224 a ;
%p A000224 end proc: # _R. J. Mathar_, Oct 10 2014
%t A000224 Length[Union[#]]& /@ Table[Mod[k^2, n], {n, 65}, {k, n}] (* _Jean-François Alcover_, Aug 30 2011 *)
%t A000224 a[2] = 2; a[n_] := a[n] = Switch[fi = FactorInteger[n], {{_, 1}}, (fi[[1, 1]] + 1)/2, {{2, _}}, 3/2 + 2^fi[[1, 2]]/6 + (-1)^(fi[[1, 2]]+1)/6, {{_, _}}, {p, k} = fi[[1]]; 3/4 + (p-1)*(-1)^(k+1)/(4*(p+1)) + p^(k+1)/(2*(p+1)), _, Times @@ Table[ a[Power @@ f], {f, fi}]]; Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Mar 09 2015 *)
%o A000224 (PARI) a(n) = local(v,i); v = vector(n,i,0); for(i=0, floor(n/2),v[i^2%n+1] = 1); sum(i=1,n,v[i]) \\ _Franklin T. Adams-Watters_, Nov 05 2006
%o A000224 (PARI) a(n)=my(f=factor(n));prod(i=1,#f[,1],if(f[i,1]==2,2^f[1,2]\6+2,f[i,1]^(f[i,2]+1)\(2*f[i,1]+2)+1)) \\ _Charles R Greathouse IV_, Jul 15 2011
%o A000224 (Haskell)
%o A000224 a000224 n = product $ zipWith f (a027748_row n) (a124010_row n) where
%o A000224 f 2 e = 2 ^ e `div` 6 + 2
%o A000224 f p e = p ^ (e + 1) `div` (2 * p + 2) + 1
%o A000224 -- _Reinhard Zumkeller_, Aug 01 2012
%o A000224 (Python)
%o A000224 from math import prod
%o A000224 from sympy import factorint
%o A000224 def A000224(n): return prod((p**(e+1)//((p+1)*(q:=1+(p==2)))>>1)+q for p, e in factorint(n).items()) # _Chai Wah Wu_, Oct 07 2024
%Y A000224 Cf. A095972, A046530 (cubic residues), A052273 (4th powers), A052274 (5th powers), A052275 (6th powers), A085310 (7th powers), A085311 (8th powers), A085312 (9th powers), A085313 (10th powers), A085314 (11th powers), A228849 (12th powers).
%K A000224 nonn,easy,nice,mult
%O A000224 1,2
%A A000224 _N. J. A. Sloane_