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%I #45 Nov 01 2022 04:59:57
%S 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,
%T 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,
%U 25,55,2,57,27,9,32,9,7,63,26,19,3,67,12,69,33,5,30,21,9,75,8
%N 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.
%C 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.
%C If n is prime, a(n) = max(1,n-4).
%D V. A. Golubev, Sur certaines fonctions multiplicatives et le problème des jumeaux. Mathesis 67 (1958), 11-20.
%D József Sándor and Borislav Crstici, Handbook of Number Theory II, Kluwer, 2004, p. 289.
%H Amiram Eldar, <a href="/A319516/b319516.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 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 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 Multiplicative with a(p^e) = p^(e-1) if p <= 5; (p-4)*p^(e-1) if p > 5.
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (49/200) * Product_{p prime >= 7} (1 - 4/p^2) = 0.1987646881... . - _Amiram Eldar_, Nov 01 2022
%e 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.
%t 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 *)
%t f[p_, e_] := If[p < 7, p^(e-1), (p-4)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 22 2020 *)
%o (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));
%o for(n=1,80,print1(phi4(n)","))
%Y Cf. similar generalizations of totient for k-tuples: A002472 (k=2), A319534 (k=3), A321029 (k=5), A321030 (k=6).
%K nonn,mult
%O 1,4
%A _Alexei Kourbatov_, Sep 21 2018
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