

A123565


a(n) is the number of positive integers k which are <= n and where k, k1 and k+1 are each coprime to n.


13



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

1,5


COMMENTS

a(p) = p3 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
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



FORMULA

Multiplicative with a(2^e) = 0 and a(p^e) = (p3)*p^(e1) for odd primes p.  Amiram Eldar, Nov 22 2020
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 (k1) 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



CROSSREFS



KEYWORD

nonn,mult


AUTHOR



EXTENSIONS



STATUS

approved



