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A123565
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a(n) = number of positive integers k which are <= n and where k, k-1 and k+1 are each coprime to n.
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1
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1, 0, 0, 0, 2, 0, 4, 0, 0, 0, 8, 0, 10, 0, 0, 0, 14, 0, 16, 0, 0, 0, 20, 0, 10, 0, 0, 0, 26, 0, 28, 0, 0, 0, 8, 0, 34, 0, 0, 0, 38, 0, 40, 0, 0, 0, 44, 0, 28, 0, 0, 0, 50, 0, 16, 0, 0, 0, 56, 0, 58, 0, 0, 0, 20, 0, 64, 0, 0, 0, 68, 0, 70, 0, 0, 0, 32, 0, 76, 0, 0, 0, 80, 0, 28, 0, 0, 0, 86, 0, 40, 0, 0
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OFFSET
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1,5
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COMMENTS
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a(p) = p-3 for any odd prime p. a(2n) = a(3n) = 0.
a(n) > 0 if and only if n is coprime to 6. - Chai Wah Wu, Aug 26 2016
Also number of pandiagonal Latin squares of order n with fixed first row. - Eduard I. Vatutin, Oct 15 2017
Multiplicative by the Chinese remainder theorem. - Andrew Howroyd, Aug 07 2018
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LINKS
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Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
C. Defant, On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions, J. Int. Seq. 18 (2015) # 15.2.1
E. I. Vatutin, Discussion about properties of diagonal Latin squares at forum.boinc.ru (in Russian)
Index entries for sequences related to Latin squares and rectangles
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EXAMPLE
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The positive integers which are both coprime to 25 and are <= 25 are 1,2,3,4,6,7,8,9,11,12,13,14,16,17,18,19,21,22,23,24. Of these integers there are 10 integers k where (k-1) and (k+1) are also coprime to 25. These integers k are 2,3,7,8,12,13,17,18,22,23. So a(25) = 10.
Example of pandiagonal Latin square of order 5:
0 1 2 3 4
2 3 4 0 1
4 0 1 2 3
1 2 3 4 0
3 4 0 1 2
Example of pandiagonal Latin square of order 7:
0 1 2 3 4 5 6
2 3 4 5 6 0 1
4 5 6 0 1 2 3
6 0 1 2 3 4 5
1 2 3 4 5 6 0
3 4 5 6 0 1 2
5 6 0 1 2 3 4
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MATHEMATICA
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f[n_] := Length[Select[Range[n], GCD[ #, n] == 1 && GCD[ # - 1, n] == 1 && GCD[ # + 1, n] == 1 &]]; Table[f[n], {n, 100}] (* Ray Chandler, Nov 19 2006 *)
Join[{1}, Table[Count[Boole[Partition[CoprimeQ[Range[n], n], 3, 1]], {1, 1, 1}], {n, 2, 100}]] (* Harvey P. Dale, Apr 09 2017 *)
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PROG
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(PARI) a(n)=if(gcd(n, 6)>1, return(0)); sum(k=1, n, gcd(k^3-k, n)==1) \\ Charles R Greathouse IV, Aug 26 2016
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CROSSREFS
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Cf. A058026.
Sequence in context: A087263 A099894 A048298 * A258701 A246160 A081120
Adjacent sequences: A123562 A123563 A123564 * A123566 A123567 A123568
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KEYWORD
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nonn,mult
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AUTHOR
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Leroy Quet, Nov 12 2006
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EXTENSIONS
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Extended by Ray Chandler, Nov 19 2006
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STATUS
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approved
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