

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.


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
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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
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 noncyclic pandiagonal Latin squares of prime orders, Journal of Discrete Algorithms, Vol. 30 (2015), pp. 7077.
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. 191192.
Eduard I. Vatutin, Enumerating cyclic Latin squares and Euler totient function calculating using them, Highperformance computing systems and technologies, 2020, Vol. 4, No. 2, pp. 4048. (in Russian)
Index entries for sequences related to Latin squares and rectangles.


FORMULA

Multiplicative with a(2^e) = 0 and a(p^e) = (p3)*p^(e1) 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 (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

(PARI) a(n)=if(gcd(n, 6)>1, return(0)); sum(k=1, n, gcd(k^3k, 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



