cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A306538 The least prime q such that Kronecker(D/q) = 1 where D runs through all negative fundamental discriminants (-A003657).

Original entry on oeis.org

7, 5, 2, 3, 3, 2, 5, 3, 2, 5, 2, 3, 2, 7, 11, 2, 5, 7, 2, 3, 3, 17, 3, 2, 2, 3, 5, 2, 13, 5, 2, 2, 3, 3, 2, 7, 3, 2, 11, 11, 2, 3, 7, 5, 5, 2, 19, 2, 3, 3, 2, 41, 3, 2, 13, 3, 2, 5, 7, 2, 7, 2, 3, 5, 3, 2, 5, 2, 3, 11, 2, 31, 13, 2, 5, 2, 3, 3, 2, 5, 3, 2, 5, 23, 2, 5, 17, 7, 2, 5, 7, 2, 3, 3
Offset: 1

Views

Author

Jianing Song, Feb 22 2019

Keywords

Comments

a(n) is the least prime that decomposes in the imaginary quadratic field with discriminant D, D = -A003657(n).
For most n, a(n) is relatively small. There are only 472 n's among [1, 3043] (there are 3043 terms in A003657 below 10000) that violate a(n) < log(A003657(n)).
Also a(n) is the smallest prime p such that the imaginary quadratic field with discriminant D = -A003657(n) can be embedded into the p-adic field Q_p. - Jianing Song, Feb 14 2021

Examples

			Let K = Q[sqrt(-177)] with D = -708 = -A003657(218), we have: 2 and 3 divides 708, (-708/5) = (-708/7) = ... = (-708/29) = -1 and (-708/31) = +1, so 2 and 3 ramify in K, 5, 7, ..., 29 remain inert in K and 31 decomposes in K, so a(218) = 31.

Crossrefs

Cf. A003657.
Similar sequences: A232931, A232932 (the least prime that remains inert); A306537, this sequence (the least prime that decomposes); A306541, A306542 (the least prime that decomposes or ramifies).

Programs

  • PARI
    b(D)=forprime(p=2, oo, if(kronecker(D, p)>0, return(p)))
for(n=1, 300, if(isfundamental(-n), print1(b(-n), ", ")))

A306537 The least prime q such that Kronecker(D/q) = 1 where D runs through all positive fundamental discriminants (A003658).

Original entry on oeis.org

2, 11, 7, 11, 3, 2, 5, 5, 3, 5, 2, 3, 3, 2, 5, 7, 5, 2, 7, 3, 2, 5, 2, 3, 13, 3, 3, 2, 7, 7, 2, 5, 5, 2, 3, 2, 7, 3, 2, 3, 3, 2, 13, 5, 2, 5, 11, 5, 3, 2, 7, 11, 3, 13, 2, 3, 3, 2, 11, 2, 7, 2, 5, 3, 2, 11, 2, 3, 5, 3, 3, 2, 5, 13, 2, 13, 2, 3, 2, 5, 2, 3, 5, 2
Offset: 1

Views

Author

Jianing Song, Feb 22 2019

Keywords

Comments

a(n) is the least prime that decomposes in the real quadratic field with discriminant D, D = A003658(n).
For most n, a(n) is relatively small. There are only 459 n's among [1, 3044] (there are 3044 terms in A003658 below 10000) that violate a(n) < log(A003658(n)).
Also a(n) is the smallest prime p such that the real quadratic field with discriminant D = A003658(n) can be embedded into the p-adic field Q_p. - Jianing Song, Feb 14 2021

Examples

			Let K = Q[sqrt(635)] with D = 2540 = A003658(774), we have: 2 and 5 divides 2540, (2540/3) = (2540/7) = ... = (2540/37) = -1 and (2540/41) = +1, so 2 and 5 ramify in K, 3, 7, ..., 37 remain inert in K and 41 decomposes in K, so a(774) = 41.

Crossrefs

Cf. A003658.
Similar sequences: A232931, A232932 (the least prime that remains inert); this sequence, A306538 (the least prime that decomposes); A306541, A306542 (the least prime that decomposes or ramifies).

Programs

  • PARI
    b(D)=forprime(p=2, oo, if(kronecker(D, p)>0, return(p)))
for(n=1, 300, if(isfundamental(n), print1(b(n), ", ")))

A306542 The least prime q such that Kronecker(D/q) >= 0 where D runs through all negative fundamental discriminants (-A003657).

Original entry on oeis.org

3, 2, 2, 2, 3, 2, 5, 2, 2, 2, 2, 3, 2, 2, 11, 2, 3, 2, 2, 2, 3, 17, 2, 2, 2, 3, 2, 2, 2, 5, 2, 2, 2, 3, 2, 5, 2, 2, 2, 3, 2, 3, 2, 2, 5, 2, 2, 2, 2, 3, 2, 41, 2, 2, 2, 3, 2, 2, 7, 2, 3, 2, 3, 5, 2, 2, 3, 2, 3, 2, 2, 2, 5, 2, 2, 2, 2, 3, 2, 5, 2, 2, 2, 3, 2, 2, 2, 7
Offset: 1

Views

Author

Jianing Song, Feb 22 2019

Keywords

Comments

a(n) is the least prime that either decomposes or ramifies in the imaginary quadratic field with discriminant D, D = -A003657(n).
The quadratic field with discriminant D = -A003657(n) has class number 1 if and only if a(n) >= (1/4)*A003657(n). If the quadratic field with discriminant D = -A003657(n) has class number 3 then a(n)^2 < (1/4)*A003657(n) < a(n)^3.
For most n, a(n) is relatively small. There are only 86 n's among [1, 3043] (there are 3043 terms in A003657 below 10000) that violate a(n) < log(A003657(n)). In fact, if we ignore the first term, the only terms among the first 3043 ones that seem unusually large are a(15) = 11 (with A003657(15) = 43), a(22) = 17 (with A003657(22) = 67), a(52) = 41 (with A003657(52) = 163), a(1147) = 23 (with A003657(1147) = 3763), a(2677) = 23 (with A003657(2677) = 8803) and a(2758) = 23 (with A003657(2758) = 9067).

Examples

			Let K = Q[sqrt(-3763)] with D = -3763 = -A003657(1147), we have: (-3763/2) = (-3763/3) = ... = (-3763/19) = -1 and (-3763/23) = +1, so 2, 3, 5, 7, 11, 13, 17 and 19 remain inert in K and 23 decomposes in K, so a(1147) = 23.

Crossrefs

Cf. A003657.
Similar sequences: A232931, A232932 (the least prime that remains inert); A306537, A306538 (the least prime that decomposes); A306541, this sequence (the least prime that decomposes or ramifies).

Programs

  • PARI
    b(D)=forprime(p=2, oo, if(kronecker(D, p)>=0, return(p)))
for(n=1, 300, if(isfundamental(-n), print1(b(-n), ", ")))
Showing 1-3 of 3 results.

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