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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

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).

REFERENCES

V. A. Golubev, Sur certaines fonctions multiplicatives et le problème des jumeaux. Mathesis 67 (1958), 11-20.

J. Sándor, B. Crstici, Handbook of Number Theory, vol.II. Kluwer, 2004, p.289.

LINKS

Table of n, a(n) for n=1..80.

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.

FORMULA

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

EXAMPLE

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.

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 *)

PROG

(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)", "))

CROSSREFS

Cf. similar generalizations of totient for k-tuples: A002472 (k=2), A319534 (k=3), A321029 (k=5), A321030 (k=6).

Sequence in context: A306565 A055068 A237498 * A015138 A157807 A100529

Adjacent sequences:  A319513 A319514 A319515 * A319517 A319518 A319519

KEYWORD

nonn,mult

AUTHOR

Alexei Kourbatov, Sep 21 2018

STATUS

approved

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Last modified June 24 13:17 EDT 2019. Contains 324325 sequences. (Running on oeis4.)