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A133755
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Number of positive integers less than n/3 that are coprime to n.
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1
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0, 0, 0, 1, 1, 1, 2, 1, 2, 2, 3, 1, 4, 2, 3, 3, 5, 2, 6, 2, 4, 4, 7, 3, 7, 4, 6, 4, 9, 2, 10, 5, 7, 6, 8, 4, 12, 6, 8, 6, 13, 4, 14, 6, 8, 8, 15, 5, 14, 6, 11, 8, 17, 6, 14, 8, 12, 10, 19, 6, 20, 10, 12, 11, 16, 6, 22, 10, 15, 8, 23, 8, 24, 12, 13, 12, 20, 8, 26, 10, 18, 14, 27, 8, 22, 14, 19, 14
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OFFSET
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1,7
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COMMENTS
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If the definition were changed to "... <= ...", the term a(3) would change from 0 to 1, but all others remain the same. Obviously a(n) is also the number of reduced fractions < 1/3 with denominator equal to n.
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LINKS
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FORMULA
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For n>3, a(n) = (eulerphi(n) + c) / 3 where the term c is nonzero if and only if eulerphi(n) is not divisible by 3. In the latter case n=3^t*p1^k1*...*pm^km where every prime pi=2 (mod 3) and t=0 or 1 and the value of c is given by: c = (-1)^(t+k1+...+km) * 2^(m-1). - Max Alekseyev, Jan 07 2007
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MATHEMATICA
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f[n_] := Block[{c = 0, k = 1, lmt = n/3}, While[k < lmt, If[ GCD[k, n] == 1, c++ ]; k++ ]; c]; Array[f, 88] (* Robert G. Wilson v, Jan 06 2008 *)
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PROG
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(PARI) vector(100, i, sum(j=1, (i-1)\3, gcd(i, j)==1))
(PARI) a(n)=sumdiv(n, d, moebius(n\d)*(d\3)) /* Max Alekseyev, Jan 07 2007 */
(PARI) { a(n) = ( eulerphi(n) + if(eulerphi(n)%3, (-1)^bigomega(n)*2^(omega(n)-1-(n%3==0))) )/3 } /* Max Alekseyev, Jan 07 2007 */
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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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