A319534 Number of integers x such that 1 <= x <= n and gcd(x,n) = gcd(x+2,n) = gcd(x+6,n) = 1.

1, 1, 1, 2, 2, 1, 4, 4, 3, 2, 8, 2, 10, 4, 2, 8, 14, 3, 16, 4, 4, 8, 20, 4, 10, 10, 9, 8, 26, 2, 28, 16, 8, 14, 8, 6, 34, 16, 10, 8, 38, 4, 40, 16, 6, 20, 44, 8, 28, 10, 14, 20, 50, 9, 16, 16, 16, 26, 56, 4, 58, 28, 12, 32, 20, 8, 64, 28, 20, 8, 68, 12, 70, 34, 10, 32, 32, 10, 76, 16

Offset: 1

Author

Alexei Kourbatov, Sep 22 2018

Keywords

• nonn
• mult

Comments

Equivalently, a(n) is the number of "admissible" residue classes modulo n which are allowed (by divisibility considerations) to contain infinitely many initial primes in prime triples (p, p+2, p+6). This sequence also gives the number of "admissible" residue classes (mod n) for initial primes p in the other type of prime triples: (p,p+4,p+6). This is a generalization of Euler's totient function (the number of residue classes modulo n containing infinitely many primes).

If n is prime, a(n) = max(1,n-3).

Examples

			All initial primes p in prime triples (p, p+2, p+6) are congruent to 5 mod 6; that is, there is only one "admissible" residue class mod 6; therefore a(6) = 1.
		

References

• V. A. Golubev, Sur certaines fonctions multiplicatives et le problème des jumeaux. Mathesis 67 (1958), 11-20.
• József Sándor and Borislav Crstici, Handbook of Number Theory II, Kluwer, 2004, p. 289.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• V. A. Golubev, A generalization of the functions phi(n) and pi(x), Časopis pro pěstování matematiky 78 (1953), 47-48.
• V. A. Golubev, Exact formulas for the number of twin primes and other generalizations of the function pi(x), Časopis pro pěstování matematiky 87 (1962), 296-305.
• Alexei Kourbatov and Marek Wolf, Predicting maximal gaps in sets of primes, arXiv preprint arXiv:1901.03785 [math.NT], 2019.

Crossrefs

Cf. similar generalizations of totient for k-tuples: A002472 (k=2), A319516 (k=4), A321029 (k=5), A321030 (k=6).

Programs

Mathematica

a[n_] := Sum[Boole[CoprimeQ[n, x] && CoprimeQ[n, x+2] && CoprimeQ[n, x+6]], {x, 1, n}]; Array[a, 80] (* Jean-François Alcover, Jan 29 2019 *)
f[p_, e_] := If[p < 5, p^(e-1), (p-3)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 22 2020 *)

PARI

phi3(n) = sum(x=1, n, (gcd(n, x)==1) && (gcd(n, x+2)==1) && (gcd(n, x+6)==1));
for(n=1, 80, print1(phi3(n)", "))

Formula

Multiplicative with a(p^e) = p^(e-1) if p = 2,3; (p-3)*p^(e-1) if p > 3.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (7/24) * Product_{p prime >= 5} (1 - 3/p^2) = 0.2196022165... . - Amiram Eldar, Nov 01 2022