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 A123565 a(n) is the number of positive integers k which are <= n and where k, k-1 and k+1 are each coprime to n. 11
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS 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 Multiplicative by the Chinese remainder theorem. - Andrew Howroyd, Aug 07 2018 From Eduard I. Vatutin, Nov 03 2020: (Start) a(n) is the number of cyclic diagonal Latin squares of order n with the first row in order. Every cyclic diagonal Latin square is a cyclic Latin square, so a(n) <= A000010(n). Every cyclic diagonal Latin square is pandiagonal, but the converse is not true. For example, for order n=13 there is a square 7 1 0 3 6 5 12 2 8 9 10 11 4 2 3 4 10 0 7 6 9 12 11 5 8 1 4 11 1 7 8 9 10 3 6 0 12 2 5 6 5 8 11 10 4 7 0 1 2 3 9 12 8 9 2 5 12 11 1 4 3 10 0 6 7 3 6 12 0 1 2 8 11 5 4 7 10 9 10 0 3 2 9 12 5 6 7 8 1 4 11 1 7 10 4 3 6 9 8 2 5 11 12 0 11 4 5 6 7 0 3 10 9 12 2 1 8 5 8 7 1 4 10 11 12 0 6 9 3 2 12 2 9 8 11 1 0 7 10 3 4 5 6 9 10 11 12 5 8 2 1 4 7 6 0 3 0 12 6 9 2 3 4 5 11 1 8 7 10 that is pandiagonal but not cyclic (Dabbaghian and Wu). (End) Schemmel's totient function of order 3 (Schemmel, 1869; Sándor and Crstici, 2004). - Amiram Eldar, Nov 22 2020 a(p) is a lower bound for cardinality of clique of MODLS for all odd prime orders p: a(p) <= A328873(p). - Eduard I. Vatutin, Apr 02 2021 REFERENCES József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 276. LINKS Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 Vahid Dabbaghian and Tiankuang Wu, Constructing non-cyclic pandiagonal Latin squares of prime orders, Journal of Discrete Algorithms, Vol. 30 (2015), pp. 70-77. Colin Defant, On Arithmetic Functions Related to Iterates of the Schemmel Totient Functions, J. Int. Seq., Vol. 18 (2015), Article # 15.2.1. Victor Schemmel, Ueber relative Primzahlen, Journal für die reine und angewandte Mathematik, Vol. 70 (1869), pp. 191-192. Eduard I. Vatutin, Enumerating cyclic Latin squares and Euler totient function calculating using them, High-performance computing systems and technologies, 2020, Vol. 4, No. 2, pp. 40-48. (in Russian) FORMULA Multiplicative with a(2^e) = 0 and a(p^e) = (p-3)*p^(e-1) for odd primes p. - Amiram Eldar, Nov 22 2020 a(2*n+1) = A338562(n) / (2*n+1)!. - Eduard I. Vatutin, Apr 02 2021 Sum_{k=1..n} a(k) ~ c * n^2, where c = Product_{p prime} (1 - 3/p^2) = 0.125486... (A206256). - Amiram Eldar, Nov 18 2022 EXAMPLE 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 a cyclic diagonal 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 a cyclic diagonal 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 MATHEMATICA 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 *) f[2, e_] := 0; f[p_, e_] := (p - 3)*p^(e - 1); a = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 22 2020 *) PROG (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 CROSSREFS Cf. A000010, A058026, A206256, A241663, A328873. Sequence in context: A087263 A099894 A048298 * A258701 A246160 A081120 Adjacent sequences: A123562 A123563 A123564 * A123566 A123567 A123568 KEYWORD nonn,mult AUTHOR Leroy Quet, Nov 12 2006 EXTENSIONS Extended by Ray Chandler, Nov 19 2006 STATUS approved

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Last modified February 8 01:34 EST 2023. Contains 360133 sequences. (Running on oeis4.)