

A319534


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


4



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
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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 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,n3).


REFERENCES

V. A. Golubev, Sur certaines fonctions multiplicatives et le problème des jumeaux. Mathesis 67 (1958), 1120.
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), 4748.
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), 296305.
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^(e1) if p = 2,3; (p3)*p^(e1) if p > 3.


EXAMPLE

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.


MAPLE

P:=proc(n) local a, j, x; a:=0; for x from 1 to n do
if add(gcd(x+j, n), j=[0, 2, 6])=3 then a:=a+1; fi; od;
a; end: seq(P(i), i=1..80); # Paolo P. Lava, Jan 29 2019


MATHEMATICA

a[n_] := Sum[Boole[CoprimeQ[n, x] && CoprimeQ[n, x+2] && CoprimeQ[n, x+6]], {x, 1, n}]; Array[a, 80] (* JeanFrançois Alcover, Jan 29 2019 *)


PROG

(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)", "))


CROSSREFS

Cf. similar generalizations of totient for ktuples: A002472 (k=2), A319516 (k=4), A321029 (k=5), A321030 (k=6).
Sequence in context: A225639 A110664 A193922 * A061436 A214095 A213948
Adjacent sequences: A319531 A319532 A319533 * A319535 A319536 A319537


KEYWORD

nonn,mult


AUTHOR

Alexei Kourbatov, Sep 22 2018


STATUS

approved



