A038540 -id:A038540 - cp's OEIS frontend

A352550 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 5.

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

Offset: 1

N. J. A. Sloane, Mar 20 2022

nonn

Don Zagier, On the Number of n-Element Modules Over the Ring of Integers in a Quadratic Number Field [Based on email to N. J. A. Sloane, March 18 2022]

Cf. A038540, A038541, A248107, A352551-A352567.

\\ Don Zagier, Mar 18 2022
PZ(D,m=20) = Z=dirmul(vector(m,n,1),vD=vector(m,n,kronecker(D,n))); v=Z; \
for(j=2,log(m)/log(2), V=v*0;for(k=1,m^(1/j),V[k^j]=Z[k]);v=dirmul(v,V)); v
PZ(5,100)

A038541 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -20.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 2, 3, 5, 1, 0, 4, 0, 2, 2, 5, 0, 5, 0, 2, 4, 0, 2, 6, 2, 0, 10, 4, 2, 2, 0, 7, 0, 0, 2, 10, 0, 0, 0, 3, 2, 4, 2, 0, 5, 2, 2, 10, 5, 2, 0, 0, 0, 10, 0, 6, 0, 2, 0, 4, 2, 0, 10, 11, 0, 0, 2, 0, 4, 2, 0, 15, 0, 0, 4, 0, 0, 0, 0, 5, 20, 2, 2, 8, 0, 2, 4, 0, 2, 5, 0, 4, 0, 2, 0, 14

Offset: 1

Paolo Dominici (pl.dm(AT)libero.it)

Equivalently, number of modules with n elements over the ring Z[sqrt(-5)].

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

Edited by M. F. Hasler, Feb 18 2022

Revised by N. J. A. Sloane, Mar 21 2022

A352567 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -19.

Original entry on oeis.org

1, 0, 0, 1, 2, 0, 2, 0, 1, 0, 2, 0, 0, 0, 0, 2, 2, 0, 1, 2, 0, 0, 2, 0, 5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 2, 0, 5, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 2, 3, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 1, 4, 0, 0, 4, 2, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 5, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0

Offset: 1

N. J. A. Sloane, Mar 21 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352566.

A352551 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 8.

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 3, 1, 0, 0, 0, 0, 2, 0, 5, 2, 1, 0, 0, 0, 0, 2, 0, 1, 0, 0, 4, 0, 0, 2, 7, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 5, 1, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 2, 2, 11, 0, 0, 0, 4, 0, 0, 2, 3, 2, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 2, 0, 0

Offset: 1

N. J. A. Sloane, Mar 20 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

A352561 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -3.

Original entry on oeis.org

1, 0, 1, 1, 0, 0, 2, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 0, 2, 0, 2, 0, 0, 0, 1, 0, 3, 2, 0, 0, 2, 0, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 5, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 3, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 1, 2, 0, 0, 2, 0, 5, 0, 0, 2, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0

Offset: 1

N. J. A. Sloane, Mar 21 2022

nonn

See A352550 for comments and PARI code.

Appears to be the same sequence as A248107.

Cf. A038540, A038541, A248107, A352550-A352567.

A352566 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -15.

Original entry on oeis.org

1, 2, 1, 5, 1, 2, 0, 10, 2, 2, 0, 5, 0, 0, 1, 20, 2, 4, 2, 5, 0, 0, 2, 10, 2, 0, 3, 0, 0, 2, 2, 36, 0, 4, 0, 10, 0, 4, 0, 10, 0, 0, 0, 0, 2, 4, 2, 20, 1, 4, 2, 0, 2, 6, 0, 0, 2, 0, 0, 5, 2, 4, 0, 65, 0, 0, 0, 10, 2, 0, 0, 20, 0, 0, 2, 10, 0, 0, 2, 20, 5, 0, 2, 0, 2, 0, 0, 0, 0, 4, 0, 10, 2, 4, 2, 36, 0, 2, 0, 10, 0, 4, 0, 0, 0, 4, 2, 15, 2, 0, 0, 0

Offset: 1

N. J. A. Sloane, Mar 21 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

A352552 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 12.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 0, 3, 2, 0, 2, 2, 2, 0, 0, 5, 0, 2, 0, 0, 0, 2, 2, 3, 1, 2, 3, 0, 0, 0, 0, 7, 2, 0, 0, 4, 2, 0, 2, 0, 0, 0, 0, 4, 0, 2, 2, 5, 1, 1, 0, 4, 0, 3, 0, 0, 0, 0, 2, 0, 2, 0, 0, 11, 0, 2, 0, 0, 2, 0, 2, 6, 2, 2, 1, 0, 0, 2, 0, 0, 5, 0, 2, 0, 0, 0, 0, 6, 0, 0, 0, 4, 0, 2, 0

Offset: 1

N. J. A. Sloane, Mar 20 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

A352553 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 13.

Original entry on oeis.org

1, 0, 2, 1, 0, 0, 0, 0, 5, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 0, 0, 0, 2, 0, 1, 0, 10, 0, 2, 0, 0, 0, 0, 0, 0, 5, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 4, 1, 0, 4, 1, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 3, 0, 0, 0, 2, 4, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 20, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 2, 0, 0, 0, 2, 10, 0, 0, 0, 0, 2, 0, 0, 2

Offset: 1

N. J. A. Sloane, Mar 20 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

A352554 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 17.

Original entry on oeis.org

1, 2, 0, 5, 0, 0, 0, 10, 1, 0, 0, 0, 2, 0, 0, 20, 1, 2, 2, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 0, 0, 36, 0, 2, 0, 5, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 2, 0, 10, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 65, 0, 0, 2, 5, 0, 0, 0, 10, 0, 0, 0, 10, 0, 0, 0, 0, 2, 0, 2, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 0, 2, 0, 5, 2, 0, 2, 20, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 1, 0, 0, 0, 0, 0, 2, 110, 0, 0, 0, 0, 0, 4, 0, 10, 2, 0, 0, 0, 0, 0, 0, 20, 0, 0, 0, 0, 2, 0

Offset: 1

N. J. A. Sloane, Mar 20 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

A352555 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 21.

Original entry on oeis.org

1, 0, 1, 1, 2, 0, 1, 0, 2, 0, 0, 1, 0, 0, 2, 2, 2, 0, 0, 2, 1, 0, 0, 0, 5, 0, 3, 1, 0, 0, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 2, 0, 2, 0, 4, 0, 2, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 3, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 2, 4, 5, 0, 2, 1, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 5, 2, 0, 0, 0, 2, 0, 0, 3, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane, Mar 20 2022

nonn

See A352550 for comments and PARI code.

Cf. A038540, A038541, A248107, A352550-A352567.

cp's OEIS Frontend

A352550 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 5.

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A038541 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -20.

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Extensions

A352567 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -19.

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A352551 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 8.

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A352561 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -3.

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A352566 a(n) = number of modules with n elements over the ring of integers in the imaginary quadratic field of discriminant -15.

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A352552 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 12.

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A352553 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 13.

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A352554 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 17.

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A352555 a(n) = number of modules with n elements over the ring of integers in the real quadratic field of discriminant 21.

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