A072911 -id:A072911 - cp's OEIS frontend

A384422 The number of prime powers (not including 1) p^e that divide n such that e is coprime to the p-adic valuation of n.

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 4, 2, 2, 2, 2, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 3, 2, 2, 1, 3, 2, 2, 2

Offset: 1

Amiram Eldar, May 28 2025

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A072911, A077761, A085548.

f[p_, e_] := EulerPhi[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecsum(apply(eulerphi, factor(n)[, 2]));

Additive with a(p^e) = phi(e), where phi is the Euler totient function (A000010).

Sum_{k=1..n} a(k) ~ n*(log(log(n)) + B + C), where B is Mertens's constant (A077761), C = Sum_{p prime} f(1/p) = 0.24136815875213146317..., and f(x) = -x + (1-x)*x*Sum_{k>=1} mu(k)*x^(k-1)/(1-x^k)^2.

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.

Original entry on oeis.org

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

Amiram Eldar, Feb 13 2024

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.

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.

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

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

a(n) = {my(f = factor(n)); prod(i = 1, #f~, sumdiv(f[i,2], d, f[i,1]^d * eulerphi(f[i,2]/d)));}

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).

cp's OEIS Frontend

A384422 The number of prime powers (not including 1) p^e that divide n such that e is coprime to the p-adic valuation of n.

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