A107505 -id:A107505 - cp's OEIS frontend

A107498 Theta series of quadratic form with Gram matrix [ 4, -1, 1, 1; -1, 10, 3, 3; 1, 3, 10, -3; 1, 3, -3, 88].

1, 0, 2, 0, 0, 4, 4, 4, 6, 0, 0, 8, 0, 2, 0, 8, 0, 0, 6, 8, 12, 8, 0, 0, 12, 0, 6, 0, 8, 0, 0, 12, 14, 8, 8, 0, 0, 4, 0, 8, 0, 16, 0, 0, 24, 16, 16, 24, 0, 0, 26, 0, 14, 0, 36, 0, 0, 20, 24, 28, 44, 0, 0, 32, 0, 12, 0, 20, 0, 0, 40, 36, 58, 16, 0, 0, 52, 0, 24

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

G.f. is theta_2 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

			G.f. = 1 + 2*q^4 + 4*q^10 + 4*q^12 + ...

Andy Huchala, Table of n, a(n) for n = 0..20000
W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Cf. A107497 - A107505, A140686.

prec := 60;
ls := [[4, -1, 1, 1], [-1, 10, 3, 3], [1, 3, 10, -3], [1, 3, -3, 88]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Name clarified and more terms from Andy Huchala, May 13 2023

A107503 Theta series of quadratic form with Gram matrix [ 8, 1, 1, -1; 1, 18, 5, 8; 1, 5, 18, 8; -1, 8, 8, 18].

Original entry on oeis.org

1, 0, 0, 0, 2, 0, 0, 0, 0, 6, 4, 0, 10, 2, 6, 0, 8, 6, 0, 0, 0, 0, 8, 10, 0, 6, 6, 14, 0, 14, 20, 0, 0, 0, 0, 10, 24, 0, 16, 8, 28, 0, 28, 8, 0, 0, 0, 0, 34, 16, 0, 18, 14, 8, 0, 18, 40, 0, 0, 0, 0, 22, 26, 0, 40, 12, 44, 0, 38, 28, 0, 0, 0, 0, 36, 38, 0, 30

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

G.f. is theta_7 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

			G.f. = 1 + 2*q^8 + 6*q^18 + 4*q^20 + ...

Andy Huchala, Table of n, a(n) for n = 0..20000
W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Cf. A107498, A107499, A107500, A107501, A107502, A107504, A107505.

prec := 90;
ls := [[8, 1, 1, -1], [1, 18, 5, 8], [1, 5, 18, 8], [-1, 8, 8, 18]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Name clarified and more terms from Andy Huchala, May 13 2023

A107504 Theta series of quadratic form with Gram matrix [ 12, -1, 5, 2; -1, 12, 5, 2; 5, 5, 14, 3; 2, 2, 3, 22].

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 4, 2, 6, 0, 0, 4, 0, 2, 0, 6, 0, 0, 12, 6, 14, 6, 0, 0, 16, 0, 6, 0, 18, 0, 0, 14, 14, 10, 16, 0, 0, 10, 0, 8, 0, 18, 0, 0, 22, 26, 22, 12, 0, 0, 28, 0, 14, 0, 34, 0, 0, 24, 26, 18, 50, 0, 0, 34, 0, 12, 0, 12, 0, 0, 40, 16, 56, 24, 0, 0, 36

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

G.f. is theta_8 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 14 2023

			G.f. = 1 + 4*q^12 + 2*q^14 + 6*q^16 + ...

Andy Huchala, Table of n, a(n) for n = 0..20000
W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Cf. A107498-A107503, A107505.

prec := 90;
ls := [[12, -1, 5, 2], [-1, 12, 5, 2], [5, 5, 14, 3], [2, 2, 3, 22]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 14 2023

Name clarified and more terms from Andy Huchala, May 14 2023

A107499 Theta series of quadratic form with Gram matrix [ 6, 2, 2, 1; 2, 18, 5, 9; 2, 5, 18, 9; 1, 9, 9, 24].

Original entry on oeis.org

1, 0, 0, 2, 0, 0, 0, 0, 0, 4, 4, 0, 8, 2, 10, 0, 10, 8, 0, 0, 0, 0, 8, 6, 0, 12, 6, 8, 0, 4, 22, 0, 0, 0, 0, 18, 32, 0, 10, 8, 22, 0, 26, 12, 0, 0, 0, 0, 36, 18, 0, 20, 14, 16, 0, 20, 34, 0, 0, 0, 0, 10, 22, 0, 42, 12, 42, 0, 44, 26, 0, 0, 0, 0, 38, 34, 0, 30

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

G.f. is theta_3 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

			G.f. = 1 + 2*q^6 + 4*q^18 + 4*q^20 + ...

Andy Huchala, Table of n, a(n) for n = 0..20000
W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Cf. A107498, A107500, A107501, A107502, A107503, A107504, A107505.

prec := 90;
ls := [[6, 2, 2, 1], [2, 18, 5, 9], [2, 5, 18, 9], [1, 9, 9, 24]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Name clarified and more terms from Andy Huchala, May 13 2023

A107500 Theta series of quadratic form with Gram matrix [ 10, 4, 4, 1; 4, 12, -1, 3; 4, -1, 12, 3; 1, 3, 3, 30].

Original entry on oeis.org

1, 0, 0, 0, 0, 2, 4, 4, 2, 0, 0, 2, 0, 2, 0, 8, 0, 0, 14, 2, 8, 10, 0, 0, 18, 0, 6, 0, 18, 0, 0, 4, 24, 16, 22, 0, 0, 14, 0, 8, 0, 10, 0, 0, 20, 22, 22, 14, 0, 0, 30, 0, 14, 0, 30, 0, 0, 24, 22, 10, 48, 0, 0, 24, 0, 12, 0, 24, 0, 0, 46, 22, 60, 12, 0, 0, 32, 0

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

G.f. is theta_4 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

			G.f. = 1 + 2*q^10 + 4*q^12 + 4*q^14 + ...

Andy Huchala, Table of n, a(n) for n = 0..20000
W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Cf. A107498, A107499, A107501, A107502, A107503, A107504, A107505.

prec := 90;
ls := [[10, 4, 4, 1], [4, 12, -1, 3], [4, -1, 12, 3], [1, 3, 3, 30]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Name clarified and more terms from Andy Huchala, May 13 2023

A107501 Theta series of quadratic form with Gram matrix [ 6, 3, 2, 2; 3, 8, 1, 1; 2, 1, 18, 5; 2, 1, 5, 44].

Original entry on oeis.org

1, 0, 0, 2, 4, 0, 0, 0, 0, 2, 6, 0, 6, 2, 6, 0, 8, 2, 0, 0, 0, 0, 14, 6, 0, 10, 6, 12, 0, 8, 20, 0, 0, 0, 0, 12, 24, 0, 22, 8, 28, 0, 26, 14, 0, 0, 0, 0, 36, 16, 0, 24, 14, 22, 0, 22, 30, 0, 0, 0, 0, 20, 34, 0, 30, 12, 36, 0, 32, 30, 0, 0, 0, 0, 26, 30, 0, 20

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

G.f. is theta_5 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

			G.f. = 1 + 2*q^6 + 4*q^8 + 2*q^18 + 6*q^20 + ...

Andy Huchala, Table of n, a(n) for n = 0..20000
W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Cf. A107498, A107499, A107500, A107502, A107503, A107504, A107505.

prec := 90;
ls := [[6, 3, 2, 2], [3, 8, 1, 1], [2, 1, 18, 5], [2, 1, 5, 44]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Name clarified and more terms from Andy Huchala, May 13 2023

A107502 Theta series of quadratic form with Gram matrix [ 4, 1, 0, -1; 1, 10, 0, 3; 0, 0, 26, 13; -1, 3, 13, 36].

Original entry on oeis.org

1, 0, 2, 0, 0, 2, 2, 0, 4, 0, 0, 2, 0, 2, 0, 6, 0, 0, 10, 8, 14, 12, 0, 0, 20, 0, 6, 0, 16, 0, 0, 8, 18, 18, 12, 0, 0, 12, 0, 8, 0, 6, 0, 0, 30, 22, 20, 10, 0, 0, 22, 0, 14, 0, 38, 0, 0, 22, 30, 18, 48, 0, 0, 30, 0, 12, 0, 22, 0, 0, 38, 16, 50, 30, 0, 0, 46, 0

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

G.f. is theta_6 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

			G.f. = 1 + 2*q^4 + 2*q^10 + 2*q^12 + ...

Andy Huchala, Table of n, a(n) for n = 0..20000
W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Cf. A107498, A107499, A107500, A107501, A107503, A107504, A107505.

prec := 90;
ls := [[4, 1, 0, -1], [1, 10, 0, 3], [0, 0, 26, 13], [-1, 3, 13, 36]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L, 48);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Name clarified and more terms from Andy Huchala, May 13 2023

A227131 Sum of divisors of n that are not divisible by 25. a(0) = 1.

Original entry on oeis.org

1, 1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 31, 18, 39, 20, 42, 32, 36, 24, 60, 6, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 57, 18, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 127, 84, 144, 68, 126, 96, 144, 72, 195, 74, 114, 24, 140

Offset: 0

Michael Somos, Jul 02 2013

			G.f. = 1 + q + 3*q^2 + 4*q^3 + 7*q^4 + 6*q^5 + 12*q^6 + 8*q^7 + 15*q^8 + 13*q^9 + ...
75 has six divisors: 1, 3, 5, 15, 25, 75, but both 25 and 75 are divisible by 25, thus not counted, and we have a(75) = 1+3+5+15 = 24. - _Antti Karttunen_, Nov 23 2017

Antti Karttunen, Table of n, a(n) for n = 0..16384
Index entries for sequences related to sums of divisors.

Cf. A000118, A004011, A008653, A028594, A028887, A096726, A107505.

A := Basis( ModularForms( Gamma0(25), 2), 66); A[1] + A[2] + 3*A[3] + 4*A[4] + 7*A[5]; /* Michael Somos, Jun 12 2014 */

a[ n_] := If[ n < 1, Boole[ n == 0], Sum[ If[ Mod[ d, 25] > 0, d, 0], {d, Divisors @ n}]];

{a(n) = if( n<1, n==0, sumdiv( n, d, if( d%25, d)))};

{a(n) = if( n<1, n==0, 1 * (sigma(n) - if( n%25==0, 25 * sigma( n / 25))))};

A = ModularForms( Gamma0(25), 2, prec=66) . basis(); A[0] + A[1] + 3*A[2] + 4*A[3] + 7*A[4];

a(n) is multiplicative with a(0) = 1, a(5^e) = 6 if e>0, a(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.
G.f. is a period 1 Fourier series which satisfies f(-1 / (25 t)) = 25 (t/i)^2 f(t) where q = exp(2 Pi i t).
G.f.: 1 + Sum_{k>0} k * x^k / (1 - x^k) - Sum_{k>0} 25 * k * x^(25*k) / (1 - x^(25*k)).
Sum_{k=1..n} a(k) ~ (2*Pi^2/25) * n^2. - Amiram Eldar, Oct 04 2022

More terms from Antti Karttunen, Nov 23 2017

A107497 Theta series of quadratic form with Gram matrix [ 2, 1, 1, 1; 1, 20, 7, 7; 1, 7, 20, 7; 1, 7, 7, 46].

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 8, 0, 8, 2, 4, 0, 10, 4, 0, 0, 0, 0, 12, 4, 0, 6, 6, 12, 0, 8, 20, 0, 0, 0, 0, 16, 22, 0, 24, 8, 24, 0, 32, 20, 0, 0, 0, 0, 36, 14, 0, 20, 14, 16, 0, 24, 32, 0, 0, 0, 0, 20, 28, 0, 30, 12, 44, 0, 24, 24, 0, 0, 0, 0, 28, 44, 0, 32

Offset: 0

N. J. A. Sloane, May 28 2005

nonn

G.f. is theta_1 in the Parry 1979 reference on page 166. This theta series is an element of the space of modular forms on Gamma_0(169) of weight 2 and dimension 21. - Andy Huchala, May 13 2023

			G.f. = 1 + 2*q^2 + 2*q^8 + 2*q^18 + ...

Andy Huchala, Table of n, a(n) for n = 0..20000
W. R. Parry, A negative result on the representation of modular forms by theta series, J. Reine Angew. Math., 310 (1979), 151-170.

Cf. A107498-A107505, A140686.

prec := 60;
ls := [[2,1,1,1],[1,20,7,7],[1,7,20,7],[1,7,7,46]];
S := Matrix(ls);
L := LatticeWithGram(S);
M := ThetaSeriesModularFormSpace(L);
B := Basis(M, prec);
T := ThetaSeries(L,44);
coeffs := [Coefficients(T)[2*i-1] : i in [1..23]];
Coefficients(&+[coeffs[i]*B[i] :i in [1..13]]+&+[coeffs[i+1]*B[i] :i in [14..19]] + coeffs[22]*B[20] + coeffs[23]*B[21]); // Andy Huchala, May 13 2023

Name clarified and more terms from Andy Huchala, May 13 2023

cp's OEIS Frontend

A107498 Theta series of quadratic form with Gram matrix [ 4, -1, 1, 1; -1, 10, 3, 3; 1, 3, 10, -3; 1, 3, -3, 88].

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A107503 Theta series of quadratic form with Gram matrix [ 8, 1, 1, -1; 1, 18, 5, 8; 1, 5, 18, 8; -1, 8, 8, 18].

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A107504 Theta series of quadratic form with Gram matrix [ 12, -1, 5, 2; -1, 12, 5, 2; 5, 5, 14, 3; 2, 2, 3, 22].

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A107499 Theta series of quadratic form with Gram matrix [ 6, 2, 2, 1; 2, 18, 5, 9; 2, 5, 18, 9; 1, 9, 9, 24].

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A107500 Theta series of quadratic form with Gram matrix [ 10, 4, 4, 1; 4, 12, -1, 3; 4, -1, 12, 3; 1, 3, 3, 30].

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A107501 Theta series of quadratic form with Gram matrix [ 6, 3, 2, 2; 3, 8, 1, 1; 2, 1, 18, 5; 2, 1, 5, 44].

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A107502 Theta series of quadratic form with Gram matrix [ 4, 1, 0, -1; 1, 10, 0, 3; 0, 0, 26, 13; -1, 3, 13, 36].

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Examples

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Magma

Extensions

A227131 Sum of divisors of n that are not divisible by 25. a(0) = 1.

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