cp's OEIS Frontend

A359099 a(n) = (1/6) * Sum_{d|n} phi(7 * d).

%I A359099 #37 Feb 16 2025 08:34:04
%S A359099 1,2,3,4,5,6,8,8,9,10,11,12,13,16,15,16,17,18,19,20,24,22,23,24,25,26,
%T A359099 27,32,29,30,31,32,33,34,40,36,37,38,39,40,41,48,43,44,45,46,47,48,57,
%U A359099 50,51,52,53,54,55,64,57,58,59,60,61,62,72,64,65,66,67,68,69,80,71,72,73,74,75,76,88,78,79
%N A359099 a(n) = (1/6) * Sum_{d|n} phi(7 * d).
%H A359099 Michael De Vlieger, <a href="/A359099/b359099.txt">Table of n, a(n) for n = 1..10000</a>
%H A359099 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotientFunction.html">Totient Function</a>.
%F A359099 G.f.: Sum_{k>=1} phi(7 * k) * x^k / (6 * (1 - x^k)).
%F A359099 G.f.: Sum_{k>=0} x^(7^k) / (1 - x^(7^k))^2.
%F A359099 From _Amiram Eldar_, Dec 17 2022: (Start)
%F A359099 Multiplicative with a(7^e) = (7^(e+1)-1)/6, and a(p^e) = p if p != 7.
%F A359099 Dirichlet g.f.: zeta(s-1)*(1+1/(7^s-1)).
%F A359099 Sum_{k=1..n} a(k) ~ (49/96) * n^2. (End)
%F A359099 From _Seiichi Manyama_, Jun 04 2024: (Start)
%F A359099 G.f. A(x) satisfies A(x) = x/(1 - x)^2 + A(x^7).
%F A359099 If n == 0 (mod 7), a(n) = n + a(n/7) otherwise a(n) = n. (End)
%t A359099 Array[DivisorSum[#, EulerPhi[7 #] &]/6 &, 79] (* _Michael De Vlieger_, Dec 16 2022 *)
%t A359099 f[p_, e_] := If[p == 7, (7^(e + 1) - 1)/6, p^e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 17 2022 *)
%o A359099 (PARI) a(n) = sumdiv(n, d, eulerphi(7*d))/6;
%o A359099 (PARI) my(N=80, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(7*k)*x^k/(1-x^k))/6)
%Y A359099 Cf. A129527, A327625, A359100, A373188.
%Y A359099 Cf. A000010, A373217.
%K A359099 nonn,mult
%O A359099 1,2
%A A359099 _Seiichi Manyama_, Dec 16 2022