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.

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

A055039 Numbers of the form 2^(2i+1)*(8j+7).

Original entry on oeis.org

14, 30, 46, 56, 62, 78, 94, 110, 120, 126, 142, 158, 174, 184, 190, 206, 222, 224, 238, 248, 254, 270, 286, 302, 312, 318, 334, 350, 366, 376, 382, 398, 414, 430, 440, 446, 462, 478, 480, 494, 504, 510, 526, 542, 558, 568, 574, 590, 606, 622
Offset: 1

Views

Author

N. J. A. Sloane, Jun 01 2000

Keywords

Comments

The numbers not of the form x^2+y^2+2z^2.
Numbers of the form 6*x^2 + 8*x^2*(2*y -1). (Steve Waterman).
These are the numbers not occurring as norms in the face-centered cubic lattice (cf. A004015).
Numbers whose base 4 representation ends in 3,2 followed by some number of zeros. - Franklin T. Adams-Watters, Dec 04 2006
Numbers k such that the k-th coefficient of eta(x)^4/eta(x^4) is 0 where eta is the Dedekind eta function. - Benoit Cloitre, Mar 15 2025
The asymptotic density of this sequence is 1/12. - Amiram Eldar, Mar 29 2025

Examples

			In base 4: 32, 132, 232, 320, 332, 1032, 1132, 1232, 1320, 1332, 2032, ...

Links

Crossrefs

Equals twice A004215. Not the same as A044075 - see A124169.
Complement of A000401.
Cf. A004015.

Programs

  • Mathematica
    Select[Range[650], Mod[# / 4^IntegerExponent[#, 4], 16] == 14 &] (* Amiram Eldar, Mar 29 2025 *)
  • Python
    from itertools import count, islice
def A055039_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(m:=(~n&n-1).bit_length())&1 and (n>>m)&7==7,count(max(startvalue,1)))
A055039_list = list(islice(A055039_gen(),30)) # Chai Wah Wu, Jul 09 2022
  • Python
    def A055039(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum(((x>>i)-7>>3)+1 for i in range(1,x.bit_length(),2))
    return bisection(f,n,n) # Chai Wah Wu, Feb 24 2025

A383784 Norms of vectors in any regular planar tiling (square or A2 lattice).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 7, 8, 9, 10, 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, 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
Offset: 1

Views

Author

C. S. Davis, May 09 2025

Keywords

Links

Crossrefs

Union of A001481 and A003136.
Complement of A383785.
Subsequence of A000401.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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