A091400 -id:A091400 - cp's OEIS frontend

A000089 Number of solutions to x^2 + 1 == 0 (mod n).

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

Offset: 1

N. J. A. Sloane

Number of elliptic points of order 2 for GAMMA_0(n).
The Dirichlet inverse, 1, -1, 0, 1, -2, 0, 0, -1, 0, 2, 0, 0, -2, 0,.. seems to equal A091400, apart from signs. - R. J. Mathar, Jul 15 2010
Shadow transform of A002522. - Michel Marcus, Jun 06 2013
a(n) != 0 iff n in A008784. - Joerg Arndt, Mar 26 2014
For n > 1, number of positive solutions to n = a^2 + b^2 such that gcd(a, b) = 1. - Haehun Yang, Mar 20 2022

			G.f. = x + x^2 + 2*x^5 + 2*x^10 + 2*x^13 + 2*x^17 + 2*x^25 + 2*x^26 + 2*x^29 + ...

Michael Baake, "Solution of the coincidence problem in dimensions d <= 4", in R. V. Moody, ed., Mathematics of Long-Range Aperiodic Order, Kluwer, 1997, pp. 9-44.
Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (2).

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..2000 from T. D. Noe)
Michael Baake, Solution of the coincidence problem in dimensions d <= 4, arxiv:math/0605222 [math.MG] (2006).
Michael Baake and Uwe Grimm, Quasicrystalline combinatorics, 2002.
Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
Steven R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 4].
N. J. A. Sloane, Transforms
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), Article 14.11.6.

Cf. A031358, A027748, A124010, A000095, A006278 (positions of records), A002654, A093582.

a000089 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f 2 e = if e == 1 then 1 else 0
   f p _ = if p `mod` 4 == 1 then 2 else 0
-- Reinhard Zumkeller, Mar 24 2012

with(numtheory); A000089 := proc (n) local i, s; if modp(n,4) = 0 then RETURN(0) fi; s := 1; for i in divisors(n) do if isprime(i) and i > 2 then s := s*(1+eval(legendre(-1,i))) fi od; s end: # Gene Ward Smith, May 22 2006

Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 3 ]==2, 0, Count[ Array[ Mod[ #^2+1, n ]&, n, 0 ], 0 ] ] ], 84 ]
a[ n_] := If[ n < 1, 0, Length @ Select[ (#^2 + 1)/n & /@ Range[n], IntegerQ]]; (* Michael Somos, Aug 15 2015 *)
a[n_] := a[n] = Product[{p, e} = pe; Which[p<3 && e==1, 1, p==2 && e>1, 0, Mod[p, 4]==1, 2, Mod[p, 4]==3, 0, True, a[p^e]], {pe, FactorInteger[n]}]; Array[a, 105] (* Jean-François Alcover, Oct 18 2018, after David W. Wilson *)

{a(n) = if( n<1, 0, sum( x=0, n-1, (x^2 + 1)%n==0))}; \\ Michael Somos, Mar 24 2012

a(n)=my(o=valuation(n,2),f);if(o>1,0,n>>=o;f=factor(n)[,1]; prod(i=1,#f,kronecker(-1,f[i])+1)) \\ Charles R Greathouse IV, Jul 08 2013

from math import prod
from sympy import primefactors
def A000089(n): return prod(1 if p==2 else 2 if p&3==1 else 0 for p in primefactors(n)) if n&3 else 0 # Chai Wah Wu, Oct 13 2024

a(n) = 0 if 4|n, else a(n) = Product_{ p | N } (1 + Legendre(-1, p) ), where we use the definition that Legendre(-1, 2) = 0, Legendre(-1, p) = 1 if p == 1 mod 4, = -1 if p == 3 mod 4. This is Shimura's definition, which is different from Maple's.
Dirichlet g.f.: (1+2^(-s))*Product (1+p^(-s))/(1-p^(-s)) (p=1 mod 4).
Multiplicative with a(p^e) = 1 if p = 2 and e = 1; 0 if p = 2 and e > 1; 2 if p == 1 (mod 4); 0 if p == 3 (mod 4). - David W. Wilson, Aug 01 2001
a(3*n) = a(4*n) = a(4*n + 3) = 0. a(4*n + 1) = A031358(n). - Michael Somos, Mar 24 2012
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/(2*Pi) = 0.477464... (A093582). - Amiram Eldar, Oct 11 2022

A091379 a(n) = Product_{ p | n } (1 + Legendre(-1,p) ).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 02 2004

Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (2) (but without the restriction that a(4k) = 0 and with a different definition of Legendre(-1,2)).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000086, A000091, A034444, A089491, A091392, A091393, A091394, A091395, A091396, A091397, A091398, A091399, A091400.

with(numtheory); A091379 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul((1+legendre(-1,t1[i][1])),i=1..nops(t1)); end;

a[n_] := Module[{t1, t2}, t1 = FactorInteger[n]; t2 = Product[(1 + KroneckerSymbol[-1, t1[[i, 1]]]), {i, 1, Length[t1]}]]; a[1] = 1;
Array[a, 105] (* Jean-François Alcover, Feb 08 2022, from Maple code *)

vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
A091379(n) = vecproduct(apply(p -> (1 + kronecker(-1,p)), factorint(n)[, 1])); \\ Antti Karttunen, Nov 18 2017

Here we use the definition that Legendre(-1, 2) = 1, Legendre(-1, p) = 1 if p == 1 mod 4, = -1 if p == 3 mod 4.
From Amiram Eldar, Oct 11 2022: (Start)
Multiplicative with a(p^e) = 0 if p == 3 (mod 4) and 2 otherwise.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/Pi = 0.954929... (A089491). (End)

A129448 Expansion of q * psi(-q) * chi(q^3)^2 * psi(-q^9) in powers of q where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Apr 16 2007

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Number 50 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = q - q^2 + q^4 - 2*q^5 - q^8 + 2*q^10 + 2*q^13 + q^16 - 2*q^17 - 2*q^20 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 44.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002175, A091400, A121363, A121444, A122856, A122865.

A := Basis( ModularForms( Gamma1(36), 1), 82); A[2] - A[3] + A[5] - 2*A[6] - A[9] + 2*A[11] + 2*A[14] + A[17] - 2*A[18]; /* Michael Somos, Jul 09 2015 */

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ 12, d] KroneckerSymbol[ -3, n/d], {d, Divisors[ n]}]]; (* Michael Somos, Jul 09 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q^3, q^6]^2 EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)] / (2 q^(5/4)), {q, 0, n}]; (* Michael Somos, Jul 09 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( 12, d) * kronecker( -3, n/d)))};

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, 0, p==2, (-1)^e, p%12>6, !(e%2), (-1)^(e * (p%12==5)) * (e+1))))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^4 * eta(x^9 + A) * eta(x^36 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2 * eta(x^18 + A)), n))};

Expansion of eta(q) * eta(q^4) * eta(q^6)^4 * eta(q^9) * eta(q^36) / (eta(q^2) * eta(q^3)^2 * eta(q^12)^2 * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ -1, 0, 1, -1, -1, -2, -1, -1, 0, 0, -1, -1, -1, 0, 1, -1, -1, -2, -1, -1, 1, 0, -1, -1, -1, 0, 0, -1, -1, -2, -1, -1, 1, 0, -1, -2, ...].
a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = 0^e, a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12), a(p^e) = e+1 if p == 1 (mod 12), a(p^e) = (-1)^e * (e+1) if p == 5 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker(12, k) * x^k/ (1 + x^k + x^(2*k)).
|a(n)| = A091400(n). a(3*n) = a(4*n + 3) = 0. a(2*n) = -a(n). a(3*n + 1) = A122865(n). a(3*n + 2) = - A122856(n). a(4*n + 1) = A121363(n). a(12*n + 1) = A002175(n). a(12*n + 5) = -2 * A121444(n).

A281453 Expansion of f(x, x) * f(x^7, x^11) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 1, 2, 2, 0, 3, 2, 0, 0, 2, 4, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 6, 0, 0, 0, 1, 4, 0, 2, 2, 0, 0, 2, 2, 4, 0, 0, 0, 0, 0, 0, 4, 2, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 2, 0, 0, 0, 3, 4, 0, 0, 2, 0, 4, 0, 0, 2, 0, 0

Offset: 0

Michael Somos, Jan 26 2017

nonn

			G.f. = 1 + 2*x + 2*x^4 + x^7 + 2*x^8 + 2*x^9 + 3*x^11 + 2*x^12 + ...
G.f. = q + 2*q^10 + 2*q^37 + q^64 + 2*q^73 + 2*q^82 + 3*q^100 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000122, A002654, A004018, A035154, A091400, A105673, A113446, A122856, A122857.
Cf. A122864, A122865, A125061, A129448, A138949, A138950, A163746, A205808, A256276.
Cf. A256280, A258034, A258228, A258256, A258278, A258292, A259761.
Cf. A281451, A281452, A281490, A281491, A281492.

a[ n_] := If[ n < 0, 0, DivisorSum[ 9 n + 1, KroneckerSymbol[ -4, #] &]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^7, x^18] QPochhammer[ -x^11, x^18] QPochhammer[ x^18], {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[# < 3, 1, # == 3, Mod[#2, 2] 2 + 1, Mod[#, 4] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger[ 9 n + 1])];

{a(n) = if( n<0, 0, sumdiv(9*n + 1, d, kronecker(-4, d)))};

{a(n) = if( n<0, 0, my(m = 9*n + 1, k, s); forstep(j=0, sqrtint(m), 3, if( issquare(m - j^2, &k) && (k%9 == 1 || k%9 == 8), s+=(j>0)+1)); s)};

{a(n) = if( n<0, 0, my(A, p, e); A = factor(9*n + 1); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p==3, -2*(-1)^e, p%4==1, e+1, 1-e%2)))};

f(a,b) = 1 + Sum_{k=1..oo} (ab)^(k(k-1)/2)*(a^k+b^k). - N. J. A. Sloane, Jan 30 2017
Euler transform of a period 36 sequence.
G.f.: (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(9*k^2 + 2*k)).
G.f.: Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(18*k-11)) * (1 + x^(18*k-7)) * (1 - x^(18*k)).
a(4*n + 2) = a(8*n + 5) = a(16*n + 3) = a(32*n + 31) = a(64*n + 55) = a(128*n + 39) = 0.
a(4*n + 3) = A281451(n). a(8*n + 1) = 2 * A281492(n). a(16*n + 7) = A281452(n). a(32*n + 15) = 2 * A281491(n). a(128*n + 103) = 2 * A281490(n).
a(n) = A122865(3*n) = A122856(6*n) = A258278(6*n) = a(64*n + 7). a(n) = -A256269(9*n + 1).
a(n) = b(9*n + 1) where b = A002654, A035154, A091400, A113446, A122864, A125061, A129448, A138950, A163746, A256276, A258228, A258256.
2 * a(n) = b(9*n + 1) where b = A105673, A122857, A258034, A259761. 2 * a(n) = - b(9*n+1) where b = A138949, A256280, A258292. 4 * a(n) = A004018(9*n + 1).
Convolution of A000122 and A205808.

cp's OEIS Frontend

A000089 Number of solutions to x^2 + 1 == 0 (mod n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

PARI

Python

Formula

A091379 a(n) = Product_{ p | n } (1 + Legendre(-1,p) ).

Views

Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A129448 Expansion of q * psi(-q) * chi(q^3)^2 * psi(-q^9) in powers of q where psi(), chi() are Ramanujan theta functions.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

PARI

Formula

A281453 Expansion of f(x, x) * f(x^7, x^11) in powers of x where f(, ) is Ramanujan's general theta function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

PARI

Formula