A002730 Number of equivalence classes of binary sequences of primitive period n.
2, 1, 2, 3, 4, 8, 8, 18, 18, 38, 28, 142, 72, 234, 360, 669, 520, 2606, 1608, 7338, 8856, 19370, 16768, 94308, 67556, 216200, 277512, 815310, 662368, 4499852, 2311468, 8465496, 13045076, 31592762, 40937592, 159769394, 103197488, 401912086
Offset: 1
References
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.
Links
Crossrefs
Cf. A002729.
Programs
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Maple
with(numtheory): E:=proc(k,L) if(L=1) then RETURN(1) else RETURN(order(k,L)) fi end; M:=proc(k,L) local s,EkL: EkL:=E(k,L): if(k>1) then s:=(k^EkL-1)/(k-1): RETURN(L*EkL/igcd(L,s)) else RETURN(L*EkL/igcd(L,EkL)) fi end; C:=proc(k,t,p) local u: RETURN(add(M(k,p/igcd(p,u*(k-1)+t))^(-1),u=0..p-1)) :end; N:=proc(p) options remember: local s,t,k: if(p=1) then RETURN(2) fi: s:=0: for t from 0 to p-1 do for k from 1 to p-1 do if igcd(p,k)=1 then s:=s+2^C(k,t,p) fi od od: RETURN(s/(p*phi(p))):end; Nprimitive:=proc(p) options remember: local d: RETURN(add(mobius(p/d)*N(d),d=divisors(p))): end; seq(Nprimitive(p),p=1..51); (Pab Ter)
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Mathematica
max = 38; m[k_, n_] := (s = 1; Do[ If[ Mod[s, n] == 0, Return[e], s = s + k^e ] , {e, 1, max}]); c[k_, t_, n_] := Sum[ m[k, n/GCD[n, u*(k-1) + t]]^(-1), {u, 0, n-1}]; (* b = A002729 *) b[n_] := b[n] = (s = 0; Do[ If[ GCD[n, k] == 1 , s = s + 2^c[k, t, n]] , {k, 1, n-1}, {t, 0, n-1}]; s / (n*EulerPhi[n]) ); b[0] = 1; b[1] = 2; a[n_] := Sum[ MoebiusMu[n/d]*b[d], {d, Divisors[n]}]; Table[a[n], {n, 1, max}] (* Jean-François Alcover, Dec 06 2011, after Maple *)
Formula
Reference gives formula.
Extensions
More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005
Comments