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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. 4
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

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)

Index entries for sequences related to Latin squares and rectangles

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

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] = 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, A241663.

Sequence in context: A087263 A099894 A048298 * A338620 A258701 A246160

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 January 25 21:23 EST 2021. Contains 340427 sequences. (Running on oeis4.)