A321029 Number of integers x such that 1 <= x <= n and gcd(x,n) = gcd(x+4,n) = gcd(x+6,n) = gcd(x+10,n) = gcd(x+12,n) = 1.

1, 1, 1, 2, 1, 1, 2, 4, 3, 1, 6, 2, 8, 2, 1, 8, 12, 3, 14, 2, 2, 6, 18, 4, 5, 8, 9, 4, 24, 1, 26, 16, 6, 12, 2, 6, 32, 14, 8, 4, 36, 2, 38, 12, 3, 18, 42, 8, 14, 5, 12, 16, 48, 9, 6, 8, 14, 24, 54, 2, 56, 26, 6, 32, 8, 6, 62, 24, 18, 2, 66, 12, 68, 32, 5, 28, 12, 8, 74, 8

Offset: 1

Author

Alexei Kourbatov, Oct 26 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 p in prime 5-tuples (p, p+4, p+6, p+10, p+12). This sequence also gives the number of "admissible" residue classes (mod n) for initial primes p in the other type of prime 5-tuples: (p, p+2, p+6, p+8, p+12). This sequence is a generalization of Euler's totient function (A000010(n), the number of residue classes modulo n containing infinitely many primes).

If n is prime, a(n) = max(1,n-5).

Examples

			All initial primes p in prime 5-tuples (p, p+4, p+6, p+10, p+12) are congruent to 7 mod 10; that is, there is only one "admissible" residue class mod 10; therefore a(10) = 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), A319516 (k=4), A321030 (k=6).

Programs

Mathematica

Table[Count[Range@ n, x_ /; Equal @@ Append[Map[GCD[# + x, n] &, {0, 4, 6, 10, 12}], 1]], {n, 80}] (* Michael De Vlieger, Nov 13 2018 *)
f[p_, e_] := If[p < 7, p^(e-1), (p-5)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 22 2020 *)

PARI

phi5(n) = sum(x=1, n, (gcd(n, x)==1) && (gcd(n, x+4)==1) && (gcd(n, x+6)==1) && (gcd(n, x+10)==1) && (gcd(n, x+12)==1));
for(n=1, 80, print1(phi5(n)", "))

Formula

Multiplicative with a(p^e) = p^(e-1) if p <= 5; (p-5)*p^(e-1) if p >= 7.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (49/200) * Product_{p prime >= 7} (1 - 5/p^2) = 0.1883521849... . - Amiram Eldar, Nov 01 2022