A319516 Number of integers x such that 1 <= x <= n and gcd(x,n) = gcd(x+2,n) = gcd(x+6,n) = gcd(x+8,n) = 1.

1, 1, 1, 2, 1, 1, 3, 4, 3, 1, 7, 2, 9, 3, 1, 8, 13, 3, 15, 2, 3, 7, 19, 4, 5, 9, 9, 6, 25, 1, 27, 16, 7, 13, 3, 6, 33, 15, 9, 4, 37, 3, 39, 14, 3, 19, 43, 8, 21, 5, 13, 18, 49, 9, 7, 12, 15, 25, 55, 2, 57, 27, 9, 32, 9, 7, 63, 26, 19, 3, 67, 12, 69, 33, 5, 30, 21, 9, 75, 8

Offset: 1

Author

Alexei Kourbatov, Sep 21 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 quadruples (p, p+2, p+6, p+8). 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-4).

Examples

			Some prime quadruples start with a prime congruent to 1 mod 4; others start with a prime congruent to 3 mod 4; that is, there are 2 "admissible" residue classes mod 4; therefore a(4)=2. All initial primes in prime quadruples are 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), A319534 (k=3), A321029 (k=5), A321030 (k=6).

Programs

Mathematica

a[n_] := Sum[Boole[CoprimeQ[n, x] && CoprimeQ[n, x+2] && CoprimeQ[n, x+6] && CoprimeQ[n, x+8]], {x, 1, n}]; Array[a, 80] (* Jean-François Alcover, Jan 29 2019 *)
f[p_, e_] := If[p < 7, p^(e-1), (p-4)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 22 2020 *)

PARI

phi4(n) = sum(x=1, n, (gcd(n,x)==1) && (gcd(n,x+2)==1) && (gcd(n,x+6)==1) && (gcd(n,x+8)==1));
for(n=1,80,print1(phi4(n)","))

Formula

Multiplicative with a(p^e) = p^(e-1) if p <= 5; (p-4)*p^(e-1) if p > 5.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (49/200) * Product_{p prime >= 7} (1 - 4/p^2) = 0.1987646881... . - Amiram Eldar, Nov 01 2022