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%I #34 Nov 01 2022 04:59:41
%S 1,1,1,2,1,1,2,4,3,1,6,2,8,2,1,8,12,3,14,2,2,6,18,4,5,8,9,4,24,1,26,
%T 16,6,12,2,6,32,14,8,4,36,2,38,12,3,18,42,8,14,5,12,16,48,9,6,8,14,24,
%U 54,2,56,26,6,32,8,6,62,24,18,2,66,12,68,32,5,28,12,8,74,8
%N 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) = 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 p in prime 5-tuples (p, p+4, p+6, p+10, p+12). This sequence also gives the number of "admissible" residue classes (mod n) for initial primes p in the other type of prime 5-tuples: (p, p+2, p+6, p+8, p+12). This sequence is a generalization of Euler's totient function (A000010(n), the number of residue classes modulo n containing infinitely many primes).
%C If n is prime, a(n) = max(1,n-5).
%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="/A321029/b321029.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="https://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-5)*p^(e-1) if p >= 7.
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (49/200) * Product_{p prime >= 7} (1 - 5/p^2) = 0.1883521849... . - _Amiram Eldar_, Nov 01 2022
%e All initial primes p in prime 5-tuples (p, p+4, p+6, p+10, p+12) are congruent to 7 mod 10; that is, there is only one "admissible" residue class mod 10; therefore a(10) = 1.
%t Table[Count[Range@ n, x_ /; Equal @@ Append[Map[GCD[# + x, n] &, {0, 4, 6, 10, 12}], 1]], {n, 80}] (* _Michael De Vlieger_, Nov 13 2018 *)
%t f[p_, e_] := If[p < 7, p^(e-1), (p-5)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 22 2020 *)
%o (PARI) phi5(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));
%o for(n=1, 80, print1(phi5(n)", "))
%Y Cf. similar generalizations of totient for k-tuples: A002472 (k=2), A319534 (k=3), A319516 (k=4), A321030 (k=6).
%K nonn,mult
%O 1,4
%A _Alexei Kourbatov_, Oct 26 2018
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