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.

User: Michel Waldschmidt

Michel Waldschmidt's wiki page.

Michel Waldschmidt has authored 5 sequences.

A301430 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers which are sums of two squares.

Original entry on oeis.org

3, 0, 2, 3, 1, 6, 1, 4, 2, 3, 5, 7, 0, 6, 5, 6, 3, 7, 9, 4, 7, 7, 6, 9, 9, 0, 0, 4, 8, 0, 1, 9, 9, 7, 1, 5, 6, 0, 2, 4, 1, 2, 7, 9, 5, 1, 8, 9, 3, 6, 9, 6, 4, 5, 4, 5, 8, 8, 6, 7, 8, 4, 1, 2, 8, 8, 8, 6, 5, 4, 4, 8, 7, 5, 2, 4, 1, 0, 5, 1, 0, 8, 9, 9, 4, 8, 7, 4, 6, 7, 8, 1, 3, 9, 7, 9, 2, 7, 2, 7, 0, 8, 5, 6, 7, 7
Offset: 0

Views

Author

Michel Waldschmidt, Mar 21 2018

Keywords

Comments

This is the decimal expansion of the number alpha such that the number of positive integers <= N which are sums of two squares and are also represented by the quadratic form x^2 + xy + y^2 is asymptotic to alpha*N*(log(N))^(-3/4).
Based on the constants Zeta(m=12,n=5,s=2) = 1.0482019036007..., Zeta(m=12,n=7,s=2) = 1.0262021468... and Zeta(m=12,n=11,s=2) = 1.01177863 ... read from arXiv:1008.2547 we have Product_{p == 5, 7, 11(mod 12)} (1-1/p^2)^(-1/2) = sqrt( Zeta(m=12,n=5,s=2) * Zeta(m=12,n=7,s=2) * Zeta(m=12,n=11,s=2) ) as a factor in the formulas. - R. J. Mathar, Feb 04 2021

Examples

			0.30231614235706563794776990048019971560241279...

Links

Crossrefs

Cf. A003136, A064533, A167135, A301429, A340552.

Programs

  • Maple
    Digits:= 1000: with(numtheory):
B:= evalf(3^(1/4)*Pi^(1/2)*log(2+sqrt(3))^(1/4)/(2^(5/4)*GAMMA(1/4))):
for t to 500 do p:=ithprime(t): if `or`(`or`(`mod`(p, 12) = 5, `mod`(p, 12) = 7), `mod`(p, 12) = 11) then B:= evalf(B/(1-1/p^2)^(1/2)) end if end do: B;
  • Mathematica
    prec := 200; B = N[(Sqrt[Pi] ((3 Log[2 + Sqrt[3]])/2)^(1/4))/(2 Gamma[1/4]), prec];
For[n = 3, n < 50000, n++, p = Prime[n];
If[Mod[p, 12] != 1, B = B / Sqrt[(1 - 1/p) (1 + 1/p)]]]
Print[B] (* Peter Luschny, Mar 23 2018 *)
(* -------------------------------------------------------------------------- *)
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[(3^(1/4)/2^(5/4)) * Pi^(1/2) * (Log[2 + Sqrt[3]])^(1/4) / Gamma[1/4] * Sqrt[Z[12, 5, 2] * Z[12, 7, 2] * Z[12, 11, 2]], digits]], 10, digits-1][[1]] (* Vaclav Kotesovec, Jan 15 2021 *)

Formula

Equals (3^(1/4)/2^(5/4)) * Pi^(1/2) * (log(2 + sqrt(3)))^(1/4) / Gamma(1/4) * Product_{p == 5, 7, 11 (mod 12), p prime} (1 - 1/p^2)^(-1/2).
One can base the definition on p(n) = A167135(n). Setting r(n) = (Product_{k=1..n} p(k)^2) / (Product_{k=1..n} (p(k)^2 - 1)) the rational sequence r(n) starts 4/3, 3/2, 25/16, 1225/768, 29645/18432, ... -> L. Then A301430 = sqrt(L)*M with M = ((arccosh(2)/6)^(1/4)*Gamma(3/4))/(2*sqrt(Pi)). - Peter Luschny, Mar 29 2018

Extensions

Offset corrected by Vaclav Kotesovec, Mar 25 2018
a(6)-a(10) from Peter Luschny, Mar 29 2018
More digits from Ettahri article added by Vaclav Kotesovec, May 12 2020
More digits from Vaclav Kotesovec, Jan 15 2021

A301429 Decimal expansion of an analog of the Landau-Ramanujan constant for Loeschian numbers.

Original entry on oeis.org

6, 3, 8, 9, 0, 9, 4, 0, 5, 4, 4, 5, 3, 4, 3, 8, 8, 2, 2, 5, 4, 9, 4, 2, 6, 7, 4, 9, 2, 8, 2, 4, 5, 0, 9, 3, 7, 5, 4, 9, 7, 5, 5, 0, 8, 0, 2, 9, 1, 2, 3, 3, 4, 5, 4, 2, 1, 6, 9, 2, 3, 6, 5, 7, 0, 8, 0, 7, 6, 3, 1, 0, 0, 2, 7, 6, 4, 9, 6, 5, 8, 2, 4, 6, 8, 9, 7, 1, 7, 9, 1, 1, 2, 5, 2, 8, 6, 6, 4, 3, 8, 8, 1, 4, 1, 6
Offset: 0

Views

Author

Michel Waldschmidt, Mar 21 2018

Keywords

Comments

This is the decimal expansion of the number alpha such that the number of positive integers <= N which are represented by the quadratic form x^2 + xy + y^2 is asymptotic to alpha*N/sqrt(log(N)).

Examples

			0.638909405445343882254942674928245093754975508...

References

  • S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, p. 99 (K3).

Links

Crossrefs

Cf. A003136, A003627, A064533, A301430, A333240.

Programs

  • Maple
    Digits:= 1000: A:= 2^(-1/2)*3^(-1/4):
for t to 40000 do p:= ithprime(t): if `mod`(p, 3) = 2 then
A:= evalf(A/(1-1/p^2)^(1/2)) end if end do: A;
# Alternative:
z := n -> Zeta(n)/Im(polylog(n, (-1)^(2/3))):
x := n -> (z(2^n)*(3^(2^n)-1)*sqrt(3)/2)^(1/2^n)/3:
evalf(sqrt(mul(x(n), n=1..8))/12^(1/4), 110); # Peter Luschny, Jan 17 2021
  • Mathematica
    digits = 106;
precision = digits + 10;
prodeuler[p_, a_, b_, expr_] := Product[If[a <= p <= b, expr, 1], {p, Prime[Range[PrimePi[a], PrimePi[b]]]}];
Lv3[s_] := prodeuler[p, 1, 2^(precision/s), 1/(1 - KroneckerSymbol[-3, p]*p^-s)] // N[#, precision]&;
Lv4[s_] := 2*Im[PolyLog[s, Exp[2*I*Pi/3]]]/Sqrt[3];
Lv[s_] := If[s >= 10000, Lv3[s], Lv4[s]];
gv[s_] := (1 - 3^(-s))*Zeta[s]/Lv[s];
pgv = Product[gv[2^n*2]^(2^-(n + 1)), {n, 0, 11}] // N[#, precision]&;
RealDigits[Sqrt[pgv]/12^(1/4), 10, digits][[1]]
(* Jean-François Alcover, Jan 12 2021, after PARI code due to Artur Jasinski *)
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
Z[m_, n_, s_] := (w = 1; sumz = 0; difz = 1; While[Abs[difz] > 10^(-digits - 5), difz = P[m, n, s*w]/w; sumz = sumz + difz; w++]; Exp[sumz]);
$MaxExtraPrecision = 1000; digits = 121; RealDigits[Chop[N[Pi * Sqrt[2] / (3^(7/4) * Sqrt[Z[3, 1, 2]]), digits]], 10, digits-1][[1]]
(* Vaclav Kotesovec, Jan 15 2021 *)
z[n_] := Zeta[n]/Im[PolyLog[n, (-1)^(2/3)]];
x[n_] := (z[2^n] (3^(2^n) - 1) Sqrt[3]/2)^(1/2^n)/3;
N[Sqrt[Product[x[n], { n, 8}]]/12^(1/4), 110] (* Peter Luschny, Jan 17 2021 *)

Formula

Equals 2^(-1/2)*3^(-1/4)*Product_{p == 2 (mod 3), p prime} (1 - p^(-2))^(-1/2).
One can base the definition on p(n) = A003627(n). Setting r(n) = (Product_{k=1..n} p(k)^2) / (Product_{k=1..n} (p(k)^2 - 1)) the rational sequence r(n) starts 4/3, 25/18, 605/432, 174845/124416, ... -> L. Then A301429 = sqrt(L)/12^(1/4). - Peter Luschny, Mar 29 2018 [This L is now A333240. - Peter Luschny, Jan 14 2021]
Equals Pi*sqrt(2) / (3^(7/4) * sqrt(A175646)). - Vaclav Kotesovec, May 12 2020
Equals 12^(-1/4)*Product_{n>=0} a(-n-2)*b(2^(n+1))^(2^(-n-2)) where a(n) = 3^(2^(n-1))*(1/2-3^(-2^(-n-1))/2)^(2^n) and b(n) = zeta(n)/Im(polylog(n, (-1)^(2/3))). - Peter Luschny, Jan 14 2021

Extensions

Offset corrected by Vaclav Kotesovec, Mar 25 2018
a(6)-a(10) from Peter Luschny, Mar 29 2018
More digits from Ettahri article added by Vaclav Kotesovec, May 12 2020
More digits from Vaclav Kotesovec, Jun 27 2020

A293654 Integers not represented by cyclotomic binary forms.

Original entry on oeis.org

1, 2, 6, 14, 15, 22, 23, 24, 30, 33, 35, 38, 42, 44, 46, 47, 51, 54, 56, 59, 60, 62, 66, 69, 70, 71, 77, 78, 83, 86, 87, 88, 92, 94, 95, 96, 99, 102, 105, 107, 110, 114, 115, 118, 119, 120, 123, 126, 131, 132, 134, 135, 138, 140, 141, 142, 143, 150
Offset: 1

Views

Author

Michel Waldschmidt, Feb 16 2018

Keywords

Comments

Possibly a supersequence of A055039. - C. S. Davis, May 10 2025

Links

Crossrefs

Complement of A296095.
For n>2, subsequence of A383785, following from Proposition 6.2 of Fouvry et al.

Programs

  • Maple
    g := 1;
for m from 1 to 1000 do
    for n from 3 to 50 do
        for x from -50 to 50 do
            for y from -50 to 50 do
                if (F[n] = m, max(abs(x), abs(y)) > 1
                then r[g] := m; m := m+1; n := 3;
                     x := -50; y := -50; g := g+1
                fi;
od; od; od; od;
for t to 519 do print(r[{t}] = r[t]) od;
s[1] := 1; s[2] := 2; g := 2;
for i from 1 to 518 do
    for j from r[i]+1 to r[i+1]-1 do
        g := g+1; s[g] := j
od; od;
for t to 481 do s[t] od;
  • Mathematica
    isA296095[n_] := If[n<3, Return[False], logn = Log[n]^1.161; K = Floor[ 5.383*logn]; M = Floor[2*(n/3)^(1/2)]; k = 3; While[True, If[k == 7, K = Ceiling[4.864*logn]; M = Ceiling[2*(n/11)^(1/4)]]; For[y = 2, y <= M, y++, p[z_] = y^EulerPhi[k]*Cyclotomic[k, z]; For[x = 1, x <= y, x++, If[n == p[x/y], Return[True]]]]; k++; If[k>K, Break[]]]; Return[False]];
Select[Range[150], !isA296095[#]&] (* Jean-François Alcover, Jun 21 2018, after Peter Luschny *)
  • Sage
    def A293654list(upto):
    return [n for n in (1..upto) if not isA296095(n)]
print(A293654list(150)) # Peter Luschny, Feb 25 2018

A299214 Number of representations of integers by cyclotomic binary forms.

Original entry on oeis.org

0, 0, 8, 16, 8, 0, 24, 4, 16, 8, 8, 12, 40, 0, 0, 40, 16, 4, 24, 8, 24, 0, 0, 0, 24, 8, 12, 24, 8, 0, 32, 8, 0, 8, 0, 16, 32, 0, 24, 8, 8, 0, 32, 0, 8, 0, 0, 12, 40, 12, 0, 32, 8, 0, 8, 0, 32, 8, 0, 0, 48, 0, 24, 40, 16, 0, 24, 8, 0, 0, 0, 4, 48, 8, 12, 24
Offset: 1

Views

Author

Michel Waldschmidt, Feb 16 2018

Keywords

Comments

a(m) is the number of solutions of the equation Phi_n(x,y) = m with n >= 3 and max{|x|,|y|} >= 2. Here the binary form Phi_n(x,y) is the homogeneous version of the cyclotomic polynomial phi_n(t).
One can prove that a(m) is always a multiple of 4.

Links

Crossrefs

The sequence of indices m with a(m) != 0 is A296095.
The sequence of indices m with a(m) = 0 is A293654.

Programs

  • Julia
    using Nemo
function countA296095(n)
    if n < 3 return 0 end
    R, x = PolynomialRing(ZZ, "x")
    K = Int(floor(5.383*log(n)^1.161)) # Bounds from
    M = Int(floor(2*sqrt(n/3)))        # Fouvry & Levesque & Waldschmidt
    N = QQ(n); count = 0
    for k in 3:K
        e = Int(eulerphi(ZZ(k)))
        c = cyclotomic(k, x)
        for m in 1:M, j in 0:M if max(j, m) > 1
            N == m^e*subst(c, QQ(j,m)) && (count += 1)
    end end end
    4*count
end
A299214list(upto) = [countA296095(n) for n in 1:upto]
print(A299214list(76)) # Peter Luschny, Feb 25 2018
  • Maple
    x := 'x'; y := 'y':
with(numtheory): for n from 3 to 1000 do
F[n] := expand(y^phi(n)*cyclotomic(n, x/y))  od:
g := 0:
for m from 1 to 1000 do
   for n from 3 to 60 do  # For the bounds see the reference.
      for x from -60 to 60 do
         for y from -60 to 60 do
            if F[n] = m and  max(abs(x), abs(y)) > 1
                then g := g+1 fi:
         od:
      od:
   od: a[m] := g: print(m, a[m]): g := 0
od:
  • Mathematica
    For[n = 3, n <= 100, n++, F[n] = Expand[y^EulerPhi[n] Cyclotomic[n, x/y]]]; g = 0; For[m = 1, m <= 100, m++, For[n = 3, n <= 60, n++, For[x = -60, x <= 60, x++, For[y = -60, y <= 60, y++, If[F[n] == m && Max[Abs[x], Abs[y] ] > 1, g = g+1]]]]; a[m] = g; Print[m, " ", a[m]]; g = 0];
Array[a, 100] (* Jean-François Alcover, Dec 01 2018, from Maple *)

A296095 Integers represented by cyclotomic binary forms.

Original entry on oeis.org

3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 25, 26, 27, 28, 29, 31, 32, 34, 36, 37, 39, 40, 41, 43, 45, 48, 49, 50, 52, 53, 55, 57, 58, 61, 63, 64, 65, 67, 68, 72, 73, 74, 75, 76, 79, 80, 81, 82, 84, 85, 89, 90, 91, 93, 97, 98, 100, 101, 103, 104, 106, 108, 109, 111, 112, 113, 116, 117, 121, 122
Offset: 1

Views

Author

Michel Waldschmidt, Feb 14 2018

Keywords

Comments

Possibly a subsequence of A000401. - C. S. Davis, May 10 2025
All terms divisible by 11 appear to be either of the form 11^2*A383784(n) for n>1 or x^4 + u*x^3*y + x^2*y^2 + u*x*y^3 + y^4 for x>y>0 and u={-1, 1}. - C. S. Davis, May 14 2025

Links

Crossrefs

Complement of A293654.
Supersequence of A383784(n) for n>3, according to Proposition 6.2 of Fouvry et al.

Programs

  • Julia
    using Nemo
function isA296095(n)
    n < 3 && return false
    R, z = PolynomialRing(ZZ, "z")
    N = QQ(n)
    # Bounds from Fouvry, Levesque and Waldschmidt
    logn = log(n)^1.161
    K = Int(floor(5.383*logn))
    M = Int(floor(2*(n/3)^(1/2)))
    k = 3
    while true
        c = cyclotomic(k, z)
        e = Int(eulerphi(ZZ(k)))
        if k == 7
            K = Int(ceil(4.864*logn))
            M = Int(ceil(2*(n/11)^(1/4)))
        end
        for y in 2:M, x in 1:y
            N == y^e*subst(c, QQ(x,y)) && return true
        end
        k += 1
        k > K && break
    end
    return false
end
A296095list(upto) = [n for n in 1:upto if isA296095(n)]
println(A296095list(2040)) # Peter Luschny, Feb 28 2018
  • Maple
    with(numtheory): for n from 3 to 1000 do F[n] := expand(y^phi(n)*cyclotomic(n, x/y)) od: for m to 1000 do for n from 3 to 50 do for x from -50 to 50 do for y from -50 to 50 do if `and`(F[n] = m, max(abs(x), abs(y)) > 1) then print(m); m := m+1; n := 3; x := -50; y := -50 end if end do end do end do end do;
  • Mathematica
    isA296095[n_]:=
If[n<3, Return[False],
logn = Log[n]^1.161;
K = Floor[5.383*logn];
M = Floor[2*(n/3)^(1/2)];
k = 3;
While[True,
   If[k==7,
      K = Ceiling[4.864*logn];
      M = Ceiling[2*(n/11)^(1/4)]
   ];
   For[y=2, y<=M, y++,
      p[z_] = y^EulerPhi[k]*Cyclotomic[k,z];
      For[x=1, x<=y, x++, If[n==p[x/y], Return[True]]]
   ];
   k++;
   If[k>K, Break[]]
];
Return[False]
];
Select[Range[122], isA296095] (* Jean-François Alcover, Feb 20 2018, translated from Peter Luschny's Sage script, updated Mar 01 2018 *)
  • Sage
    def isA296095(n):
    if n < 3: return False
    logn = log(n)^1.161
    K = floor(5.383*logn)
    M = floor(2*(n/3)^(1/2))
    k = 3
    while True:
        if k == 7:
            K = ceil(4.864*logn)
            M = ceil(2*(n/11)^(1/4))
        for y in (2..M):
            p = y^euler_phi(k)*cyclotomic_polynomial(k)
            for x in (1..y):
                if n == p(x/y): return True
        k += 1
        if k > K: break
    return False
def A296095list(upto):
    return [n for n in (1..upto) if isA296095(n)]
print(A296095list(122)) # Peter Luschny, Feb 28 2018

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