A293654 Integers not represented by cyclotomic binary forms.

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

Author

Michel Waldschmidt, Feb 16 2018

Keywords

• nonn

Comments

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

Links

• Michel Waldschmidt, Table of n, a(n) for n = 1..249
• Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

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