A321030 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) = gcd(x+16,n) = 1.

1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 5, 2, 7, 1, 1, 8, 11, 3, 13, 2, 1, 5, 17, 4, 5, 7, 9, 2, 23, 1, 25, 16, 5, 11, 1, 6, 31, 13, 7, 4, 35, 1, 37, 10, 3, 17, 41, 8, 7, 5, 11, 14, 47, 9, 5, 4, 13, 23, 53, 2, 55, 25, 3, 32, 7, 5, 61, 22, 17, 1, 65, 12, 67, 31, 5, 26, 5, 7, 73, 8

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 6-tuples (p, p+4, p+6, p+10, p+12, p+16). This 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-6).

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.

Formula

Multiplicative with a(p^e) = p^(e-1) if p <= 7; (p-6)*p^(e-1) if p > 7.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (43/200) * Product_{p prime >= 11} (1 - 6/p^2) = 0.1783680033... . - Amiram Eldar, Nov 01 2022

Example

All initial primes p in prime 6-tuples (p, p+4, p+6, p+10, p+12, p+16) are congruent to 7 mod 10; that is, there is only one "admissible" residue class mod 10; therefore a(10) = 1.

Mathematica

Table[Count[Range@ n, x_ /; Equal @@ Append[Map[GCD[# + x, n] &, {0, 4, 6, 10, 12, 16}], 1]], {n, 80}] (* Michael De Vlieger, Nov 13 2018 *)
f[p_, e_] := If[p <= 7, p^(e-1), (p-6)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 22 2020 *)

Prog

(PARI) phi6(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) && (gcd(n, x+16)==1));
for(n=1, 80, print1(phi6(n)", "))

Crossrefs

Cf. similar generalizations of totient for k-tuples: A002472 (k=2), A319534 (k=3), A319516 (k=4), A321029 (k=5).

Sequence in context: A322020 A294895 A285328 * A290529 A266349 A219094

Adjacent sequences: A321027 A321028 A321029 * A321031 A321032 A321033

Keyword

nonn,mult

Author

Alexei Kourbatov, Oct 26 2018

Status

approved