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A369319 a(n) is the sum of the greatest common exponential divisor of n and k over the positive numbers k that do not exceed n.

%I A369319 #46 May 30 2025 08:02:39
%S A369319 1,2,3,6,5,6,7,12,12,10,11,18,13,14,15,24,17,24,19,30,21,22,23,36,30,
%T A369319 26,33,42,29,30,31,40,33,34,35,72,37,38,39,60,41,42,43,66,60,46,47,72,
%U A369319 56,60,51,78,53,66,55,84,57,58,59,90,61,62,84,84,65,66,67
%N A369319 a(n) is the sum of the greatest common exponential divisor of n and k over the positive numbers k that do not exceed n.
%C A369319 First differs from A336465 at n = 27.
%C A369319 The sum is restricted to numbers k that have a common exponential divisor with n, i.e., numbers k with rad(k) = rad(n), where rad is the squarefree kernel function (A007947).
%C A369319 Analogous to Pillai's arithmetical function (A018804), with exponential divisors instead of divisors.
%H A369319 Amiram Eldar, <a href="/A369319/b369319.txt">Table of n, a(n) for n = 1..10000</a>
%H A369319 László Tóth, <a href="https://ac.inf.elte.hu/Vol_024_2004/doi/285_24.pdf">On certain arithmetic functions involving exponential divisors</a>, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-294; <a href="https://arxiv.org/abs/math/0610274">arXiv preprint</a>, arXiv:math/0610274v2 [math.NT], 2006-2009.
%H A369319 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/e-Divisor.html">e-Divisor</a>.
%F A369319 a(n) = Sum_{k=1..n, rad(k) = rad(n)} (n, k)_(e), where (n, k)_(e) = Product_{p|n} p^gcd(v_p(n), v_p(k)), and v_p(n) is the p-adic valuation of n (the exponent of the highest power of p that divides n).
%F A369319 Multiplicative with a(p^e) = Sum_{k=1..e} p^gcd(e, k) = Sum_{d|e} p^d * phi(e/d), where phi is the Euler totient function (A000010).
%F A369319 Dirichlet g.f.: (zeta(s-1)*zeta(2*s-1)/zeta(3*s-2)) * Product_{p prime} (1 + ((p^(s-1)-1)*(p^(2*s-1)-1)/(p^(3*s-2)-1)) * Sum_{k>=3} phi(k)/(p^(k*s-1)-1)).
%F A369319 Sum_{k=1..n} a(k) = c * n^2 / 2 + O(n * log(n)^(5/3)), where c = Product_{p prime} (1 + Sum_{k>=2} (a(p^k) - p*a(p^(k-1)))/p^(2*k)) = 1.16509457249412700814... .
%F A369319 Lim sup_{n->oo} a(n)/(n*log(log(n))) = 6 * exp(gamma)/Pi^2 (A335004).
%t A369319 f[p_, e_] := DivisorSum[e, p^#*EulerPhi[e/#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A369319 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, sumdiv(f[i,2], d, f[i,1]^d * eulerphi(f[i,2]/d)));}
%Y A369319 Cf. A000010, A007947, A018804, A072911, A308056, A335004, A336465.
%K A369319 nonn,easy,mult
%O A369319 1,2
%A A369319 _Amiram Eldar_, Feb 13 2024