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.

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, 26, 33, 42, 29, 30, 31, 40, 33, 34, 35, 72, 37, 38, 39, 60, 41, 42, 43, 66, 60, 46, 47, 72, 56, 60, 51, 78, 53, 66, 55, 84, 57, 58, 59, 90, 61, 62, 84, 84, 65, 66, 67

Offset: 1

Author

Amiram Eldar, Feb 13 2024

Keywords

• nonn
• easy
• mult

Comments

First differs from A336465 at n = 27.

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).

Analogous to Pillai's arithmetical function (A018804), with exponential divisors instead of divisors.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 24 (2004), pp. 285-294; arXiv preprint, arXiv:math/0610274v2 [math.NT], 2006-2009.
• Eric Weisstein's World of Mathematics, e-Divisor.

Crossrefs

Cf. A000010, A007947, A018804, A072911, A308056, A335004, A336465.

Programs

Mathematica

f[p_, e_] := DivisorSum[e, p^#*EulerPhi[e/#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

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)));}

Formula

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).

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).

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)).

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... .

Lim sup_{n->oo} a(n)/(n*log(log(n))) = 6 * exp(gamma)/Pi^2 (A335004).