cp's OEIS Frontend

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.

%I A321030 #27 Nov 01 2022 04:52:17
%S A321030 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,
%T A321030 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,
%U A321030 53,2,55,25,3,32,7,5,61,22,17,1,65,12,67,31,5,26,5,7,73,8
%N 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.
%C A321030 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).
%C A321030 If n is prime, a(n) = max(1,n-6).
%D A321030 V. A. Golubev, Sur certaines fonctions multiplicatives et le problème des jumeaux. Mathesis 67 (1958), 11-20.
%D A321030 József Sándor and Borislav Crstici, Handbook of Number Theory II, Kluwer, 2004, p. 289.
%H A321030 Amiram Eldar, <a href="/A321030/b321030.txt">Table of n, a(n) for n = 1..10000</a>
%H A321030 V. A. Golubev, <a href="http://dml.cz/dmlcz/117061">A generalization of the functions phi(n) and pi(x)</a>. Časopis pro pěstování matematiky 78 (1953), 47-48.
%H A321030 V. A. Golubev, <a href="http://dml.cz/dmlcz/117442">Exact formulas for the number of twin primes and other generalizations of the function pi(x)</a>, Časopis pro pěstování matematiky 87 (1962), 296-305.
%H A321030 Alexei Kourbatov and Marek Wolf, <a href="http://arxiv.org/abs/1901.03785">Predicting maximal gaps in sets of primes</a>, arXiv preprint arXiv:1901.03785 [math.NT], 2019.
%F A321030 Multiplicative with a(p^e) = p^(e-1) if p <= 7; (p-6)*p^(e-1) if p > 7.
%F A321030 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
%e A321030 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.
%t A321030 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 *)
%t A321030 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 *)
%o A321030 (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));
%o A321030 for(n=1, 80, print1(phi6(n)", "))
%Y A321030 Cf. similar generalizations of totient for k-tuples: A002472 (k=2), A319534 (k=3), A319516 (k=4), A321029 (k=5).
%K A321030 nonn,mult
%O A321030 1,4
%A A321030 _Alexei Kourbatov_, Oct 26 2018