Search: a086669 -id:a086669
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A003658
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Fundamental discriminants of real quadratic fields; indices of primitive positive Dirichlet L-series.
(Formerly M3776)
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+10
38
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1, 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, 65, 69, 73, 76, 77, 85, 88, 89, 92, 93, 97, 101, 104, 105, 109, 113, 120, 124, 129, 133, 136, 137, 140, 141, 145, 149, 152, 156, 157, 161, 165, 168, 172, 173, 177, 181, 184, 185, 188, 193, 197
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OFFSET
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1,2
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COMMENTS
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All the prime numbers in the set of positive fundamental discriminants are Pythagorean primes (A002144). - Paul Muljadi, Mar 28 2008
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REFERENCES
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Henri Cohen, A Course in Computational Algebraic Number Theory, Springer, 1993, pp. 515-519.
M. Pohst and Zassenhaus, Algorithmic Algebraic Number Theory, Cambridge Univ. Press, 1989, page 432.
Paulo Ribenboim, Algebraic Numbers, Wiley, NY, 1972, p. 97.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Squarefree numbers (multiplied by 4 if not == 1 (mod 4)).
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MATHEMATICA
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fundamentalDiscriminantQ[d_] := Module[{m, mod = Mod[d, 4]}, If[mod > 1, Return[False]]; If[mod == 1, Return[SquareFreeQ[d] && d != 1]]; m = d/4; Return[SquareFreeQ[m] && Mod[m, 4] > 1]; ]; Join[{1}, Select[Range[200], fundamentalDiscriminantQ]] (* Jean-François Alcover, Nov 02 2011, after Eric W. Weisstein *)
max = 200; Drop[Select[Union[Table[Abs[MoebiusMu[n]] * n * 4^Boole[Not[Mod[n, 4] == 1]], {n, max}]], # < max &], 1] (* Alonso del Arte, Apr 02 2014 *)
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PROG
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(PARI) v=[]; for(n=1, 500, if(isfundamental(n), v=concat(v, n))); v
(PARI) list(lim)=my(v=List()); forsquarefree(n=1, lim\4, listput(v, if(n[1]%4==1, n[1], 4*n[1]))); forsquarefree(n=lim\4+1, lim\1, if(n[1]%4==1, listput(v, n[1]))); Set(v) \\ Charles R Greathouse IV, Jan 21 2022
(Sage)
def is_fundamental(d):
r = d % 4
if r > 1 : return False
if r == 1: return (d != 1) and is_squarefree(d)
q = d // 4
return is_squarefree(q) and (q % 4 > 1)
[1] + [n for n in (1..200) if is_fundamental(n)] # Peter Luschny, Oct 15 2018
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A193140
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Number of isonemal satins of exact period n.
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+10
6
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0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 1, 0, 0, 3, 0, 1, 1, 1, 1, 0, 1, 3, 1, 1, 0, 3, 1, 0, 1, 1, 3, 1, 0, 1, 1, 1, 0, 3, 1, 1, 1, 1, 1, 1, 0, 3, 0, 1, 0, 3, 3, 0, 1, 3, 1, 1, 1, 1, 1, 0, 1, 3, 1, 0, 1, 1, 1, 1, 0, 3, 3, 1, 0, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 0, 1, 7
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OFFSET
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2,23
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COMMENTS
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On page 153 of Grünbaum and Shephard (1980) is Table 3 which is a list of all the (n,s)-satins with n<=100. - Michael Somos, Dec 05 2014
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REFERENCES
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B. Grünbaum and G. C. Shephard, The geometry of fabrics, pp. 77-98 of F. C. Holroyd and R. J. Wilson, editors, Geometrical Combinatorics. Pitman, Boston, 1984.
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LINKS
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FORMULA
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MAPLE
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U:=proc(n) local j, p3, i, t1, t2, al, even;
t1:=ifactors(n)[2];
t2:=nops(t1);
if (n mod 2) = 0 then even:=1; al:=t1[1][2]; else even:=0; al:=0; fi;
j:=t2-even;
p3:=0;
for i from 1 to t2 do if t1[i][1] mod 4 = 3 then p3:=1; fi; od:
if (al >= 2) or (p3=1) then RETURN(0) else RETURN(2^(j-1)); fi;
end;
V:=proc(n) local j, i, t1, t2, al, even;
t1:=ifactors(n)[2];
t2:=nops(t1);
if (n mod 2) = 0 then even:=1; al:=t1[1][2]; else even:=0; al:=0; fi;
j:=t2-even;
if (al <= 1) then RETURN(2^(j-1)-1); fi;
if (al = 2) then RETURN(2^j-1); fi;
if (al >= 3) then RETURN(2^(j+1)-1); fi;
end;
[seq(U(n)+V(n), n=3..120)];
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MATHEMATICA
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a[n_] := 2^With[{f = FactorInteger[n]}, Length@f - If[
f[[1, 1]] == 2 && f[[1, 2]] > 1,
Boole[f[[1, 2]] == 2],
Boole[f[[1, 1]] == 2] + Boole[AnyTrue[f[[;; , 1]], Mod[#, 4] == 3 &]]
]] - 1;
Table[a[n], {n, 2, 100}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A290098
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Characteristic function for A003658 (fundamental discriminants of real quadratic fields).
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+10
3
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1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1
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OFFSET
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1
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LINKS
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FORMULA
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Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/Pi^2 = 0.303963... (A104141). - Amiram Eldar, Jan 14 2024
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MATHEMATICA
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Block[{nn = 105, t}, t = ConstantArray[0, nn]; ReplacePart[t, Map[# -> 1 &, Select[Range@ nn, NumberFieldDiscriminant@ Sqrt@ # == # &]]]] (* Michael De Vlieger, Aug 23 2017 *)
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PROG
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(PARI) A290098(n) = isfundamental(n);
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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