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 1-6 of 6 results.

A100976 Number of all extensions over Q_2 with degree n in the algebraic closure of Q_2.

Original entry on oeis.org

1, 7, 4, 107, 6, 124, 8, 6835, 13, 762, 12, 31724, 14, 4088, 24, 6999011, 18, 26611, 20, 3121122, 32, 98292, 24, 519765964, 31, 458738, 40, 267911128, 30, 3145704, 32, 1834748739523, 48, 9437166, 48, 27903655871, 38, 41943020, 56
Offset: 1

Views

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004

Keywords

Examples

			a(2)=7: There are 6 ramified extensions with minimal polynomials x^2+2, x^2-2, x^2+6, x^2-6, x^2+2x+2, x^2+2x+6 and one unramified x^2+x+1.

References

  • M. Krasner, Le nombre des surcorps primitifs d'un degré donné et le nombre des surcorps métagaloisiens d'un degré donné d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Académie des Sciences, Paris 254, 255, 1962.

Crossrefs

Cf. A100977, A100978, A100979, A100980, A100981, A100983, A100984, A100985, A100986.

Programs

  • Maple
    p:=2; eps:=proc()local p,s,i,sum; p:=args[1]; s:=args[2]; if s=-1 then return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local p,n; p:=args[1]; n:=args[2]; return igcd(n,p^n); end: qpart:=proc() local p,n; p:=args[1]; n:=args[2]; return n/igcd(n,p^n); end: logp:=proc() local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; m:=logp(p, ppart(p,n)); h:=qpart(p,n); for s from 0 to m do summe:=summe+(p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=sigma(h)*summe;

Formula

a(n)=(sum_{d|h}d)*(sum_{s=0}^m (p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=2, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)

A100977 Number of all extensions over Q_3 with degree n in the algebraic closure of Q_3.

Original entry on oeis.org

1, 3, 22, 7, 6, 228, 8, 15, 5323, 18, 12, 5068, 14, 24, 13092, 31, 18, 1495839, 20, 42, 157424, 36, 24, 885660, 31, 42, 942953404, 56, 30, 9565848, 32, 63, 19131816, 54, 48, 24240086731, 38, 60, 200884628, 90, 42, 1033121184, 44, 84
Offset: 1

Views

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004

Keywords

Examples

			a(2)=3 There are 2 ramified extensions with minimal polynomials x^2+3, x^2-3 and one unramified x^2+2*x+2.

References

  • M. Krasner, Le nombre des surcorps primitifs d'un degré donné et le nombre des surcorps métagaloisiens d'un degré donné d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Académie des Sciences, Paris 254, 255, 1962.

Crossrefs

Cf. A100976, A100978, A100979, A100980, A100981, A100983, A100984, A100985, A100986.

Programs

  • Maple
    p:=3; eps:=proc()local p,s,i,sum; p:=args[1]; s:=args[2]; if s=-1 then return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local p,n; p:=args[1]; n:=args[2]; return igcd(n,p^n); end: qpart:=proc() local p,n; p:=args[1]; n:=args[2]; return n/igcd(n,p^n); end: logp:=proc() local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; m:=logp(p, ppart(p,n)); h:=qpart(p,n); for s from 0 to m do summe:=summe+(p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=sigma(h)*summe;

Formula

a(n)=(sum_{d|h}d)*(sum_{s=0}^m (p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=3, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)

A100978 Number of all extensions over Q_5 with degree n in the algebraic closure of Q_5.

Original entry on oeis.org

1, 3, 4, 7, 106, 12, 8, 15, 13, 1818, 12, 28, 14, 24, 12424, 31, 18, 39, 20, 109242, 32, 36, 24, 60, 8281131, 42, 40, 56, 30, 4687272, 32, 63, 48, 54, 15624848, 91, 38, 60, 56, 146484090, 42, 96, 44, 84, 634765378, 72, 48, 124, 57
Offset: 1

Views

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 24 2004

Keywords

Examples

			a(2)=3 There are 2 ramified extensions with minimal polynomials x^2-5, x^2-10 and one unramified x^2+4*x+2.

References

  • M. Krasner, Le nombre des surcorps primitifs d'un degre donne et le nombre des surcorps metagaloisiens d'un degre donne d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Academie des Sciences, Paris 254, 255, 1962

Crossrefs

Cf. A100976, A100977, A100979, A100980, A100981, A100983, A100984, A100985, A100986.

Programs

  • Maple
    p:=5; eps:=proc()local p,s,i,sum; p:=args[1]; s:=args[2]; if s=-1 then return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local p,n; p:=args[1]; n:=args[2]; return igcd(n,p^n); end: qpart:=proc() local p,n; p:=args[1]; n:=args[2]; return n/igcd(n,p^n); end: logp:=proc() local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; m:=logp(p, ppart(p,n)); h:=qpart(p,n); for s from 0 to m do summe:=summe+(p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=sigma(h)*summe;

Formula

a(n)=(sum_{d|h}d)*(sum_{s=0}^m (p^(m+s+1)-p^(2*s))/(p-1)*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=5, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)

A100979 Number of totally ramified extensions over Q_2 with degree n in the algebraic closure of Q_2.

Original entry on oeis.org

1, 6, 3, 92, 5, 90, 7, 5880, 9, 630, 11, 23028, 13, 3570, 15, 6021104, 17, 18414, 19, 2580460, 21, 90090, 23, 377290728, 25, 425958, 27, 233963492, 29, 1966050, 31, 1578396286944, 33, 8912862, 35, 19308478428, 37, 39845850, 39, 108
Offset: 1

Views

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 25 2004

Keywords

Examples

			a(2)=6 There are 6 ramified extensions with minimal polynomials x^2+2, x^2-2, x^2+6, x^2-6, x^2+2x+2, x^2+2x+6, there is another one by x^2+x+1, but this is unramified.

References

  • M. Krasner, Le nombre des surcorps primitifs d'un degre donne et le nombre des surcorps metagaloisiens d'un degre donne d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Academie des Sciences, Paris 254, 255, 1962

Crossrefs

Cf. A100976, A100977, A100978, A100980, A100981, A100983, A100984, A100985, A100986.

Programs

  • Maple
    p:=2; eps:=proc()local p,s,i,sum; p:=args[1]; s:=args[2]; if s=-1 then return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local p,n; p:=args[1]; n:=args[2]; return igcd(n,p^n); end: qpart:=proc() local p,n; p:=args[1]; n:=args[2]; return n/igcd(n,p^n); end: logp:=proc() local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; m:=logp(p, ppart(p,n)); h:=qpart(p,n); for s from 0 to m do summe:=summe+(p^s*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=n*summe;

Formula

a(n)=n*(sum_{s=0}^m p^s*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=2, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)

A100980 Number of totally ramified extensions over Q_3 with degree n in the algebraic closure of Q_3.

Original entry on oeis.org

1, 2, 21, 4, 5, 150, 7, 8, 5085, 10, 11, 2892, 13, 14, 10905, 16, 17, 984114, 19, 20, 137739, 22, 23, 472344, 25, 26, 900792441, 28, 29, 5314350, 31, 32, 17537487, 34, 35, 13832346276, 37, 38, 186535713, 40, 41, 602654010, 43, 44, 1408273477425
Offset: 1

Views

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 25 2004

Keywords

Examples

			a(4)=4 There are 4 totally ramified extensions both with Galoisgroup D_8, so 2 of them are isomorphic to Q_3[x]/(x^4+3) and two of them are isomorphic to Q_3[x]/(x^4-3)

References

  • M. Krasner, Le nombre des surcorps primitifs d'un degre donne et le nombre des surcorps metagaloisiens d'un degre donne d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Academie des Sciences, Paris 254, 255, 1962

Crossrefs

Cf. A100976, A100977, A100978, A100979, A100981, A100983, A100984, A100985, A100986.

Programs

  • Maple
    p:=3; eps:=proc()local p,s,i,sum; p:=args[1]; s:=args[2]; if s=-1 then return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local p,n; p:=args[1]; n:=args[2]; return igcd(n,p^n); end: qpart:=proc() local p,n; p:=args[1]; n:=args[2]; return n/igcd(n,p^n); end: logp:=proc() local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; m:=logp(p, ppart(p,n)); h:=qpart(p,n); for s from 0 to m do summe:=summe+(p^s*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=n*summe;

Formula

a(n)=n*(sum_{s=0}^m p^s*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=3, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)

A100981 Number of totally ramified extensions over Q_5 with degree n in the algebraic closure of Q_5.

Original entry on oeis.org

1, 2, 3, 4, 105, 6, 7, 8, 9, 1210, 11, 12, 13, 14, 9315, 16, 17, 18, 19, 62420, 21, 22, 23, 24, 8203025, 26, 27, 28, 29, 2343630, 31, 32, 33, 34, 13671735, 36, 37, 38, 39, 78124840, 41, 42, 43, 44, 439452945, 46, 47, 48, 49, 295410156050, 51
Offset: 1

Views

Author

Volker Schmitt (clamsi(AT)gmx.net), Nov 25 2004

Keywords

Examples

			a(3)=3: there is one totally ramified extension with Galois group S_3, so there are 3 totally ramified extensions in the algebraic closure all isomorphic to Q_5[x]/(x^3+5)

References

  • M. Krasner, Le nombre des surcorps primitifs d'un degre donne et le nombre des surcorps metagaloisiens d'un degre donne d'un corps de nombres p-adiques. Comptes Rendus Hebdomadaires, Academie des Sciences, Paris 254, 255, 1962

Crossrefs

Cf. A100976, A100977, A100978, A100979, A100980, A100983, A100984, A100985, A100986.

Programs

  • Maple
    p:=5; eps:=proc()local p,s,i,sum; p:=args[1]; s:=args[2]; if s=-1 then return -infinity; fi; if s=0 then return 0; fi; sum:=0; for i from 1 to s do sum:=sum+1/p^i; od; return sum; end: ppart:=proc() local p,n; p:=args[1]; n:=args[2]; return igcd(n,p^n); end: qpart:=proc() local p,n; p:=args[1]; n:=args[2]; return n/igcd(n,p^n); end: logp:=proc() local p, pp; p:=args[1]; pp:=args[2]; if op(ifactors(pp))[2]=[] then return 0; else return op(op(ifactors(pp))[2])[2]; fi; end: summe:=0; m:=logp(p, ppart(p,n)); h:=qpart(p,n); for s from 0 to m do summe:=summe+(p^s*(p^(eps(p,s)*n)-p^(eps(p,s-1)*n)); od; a(n):=n*summe;

Formula

a(n)=n*(sum_{s=0}^m p^s*(p^(eps(s)*n)-p^(eps(s-1)*n))), where p=5, n=h*p^m, with gcd(h, p)=1, eps(-1)=-infinity, eps(0)=0 and eps(s)=sum_{i=1 to s} 1/(p^i)
Showing 1-6 of 6 results.

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