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A293654 Integers not represented by cyclotomic binary forms.

%I A293654 #57 May 19 2025 20:12:11
%S A293654 1,2,6,14,15,22,23,24,30,33,35,38,42,44,46,47,51,54,56,59,60,62,66,69,
%T A293654 70,71,77,78,83,86,87,88,92,94,95,96,99,102,105,107,110,114,115,118,
%U A293654 119,120,123,126,131,132,134,135,138,140,141,142,143,150
%N A293654 Integers not represented by cyclotomic binary forms.
%C A293654 Possibly a supersequence of A055039. - _C. S. Davis_, May 10 2025
%H A293654 Michel Waldschmidt, <a href="/A293654/b293654.txt">Table of n, a(n) for n = 1..249</a>
%H A293654 Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, <a href="https://arxiv.org/abs/1712.09019">Representation of integers by cyclotomic binary forms</a>, arXiv:1712.09019 [math.NT], 2017.
%p A293654 g := 1;
%p A293654 for m from 1 to 1000 do
%p A293654     for n from 3 to 50 do
%p A293654         for x from -50 to 50 do
%p A293654             for y from -50 to 50 do
%p A293654                 if (F[n] = m, max(abs(x), abs(y)) > 1
%p A293654                 then r[g] := m; m := m+1; n := 3;
%p A293654                      x := -50; y := -50; g := g+1
%p A293654                 fi;
%p A293654 od; od; od; od;
%p A293654 for t to 519 do print(r[{t}] = r[t]) od;
%p A293654 s[1] := 1; s[2] := 2; g := 2;
%p A293654 for i from 1 to 518 do
%p A293654     for j from r[i]+1 to r[i+1]-1 do
%p A293654         g := g+1; s[g] := j
%p A293654 od; od;
%p A293654 for t to 481 do s[t] od;
%t A293654 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]];
%t A293654 Select[Range[150], !isA296095[#]&] (* _Jean-François Alcover_, Jun 21 2018, after _Peter Luschny_ *)
%o A293654 (Sage)
%o A293654 def A293654list(upto):
%o A293654     return [n for n in (1..upto) if not isA296095(n)]
%o A293654 print(A293654list(150)) # _Peter Luschny_, Feb 25 2018
%Y A293654 Complement of A296095.
%Y A293654 For n>2, subsequence of A383785, following from Proposition 6.2 of Fouvry et al.
%K A293654 nonn
%O A293654 1,2
%A A293654 _Michel Waldschmidt_, Feb 16 2018