A363924 a(n) = number of k <= m such that rad(k) | m, where m = A005117(n) and rad(n) = A007947(n).

1, 2, 2, 2, 5, 2, 6, 2, 2, 6, 5, 2, 2, 5, 7, 2, 7, 2, 18, 2, 6, 8, 5, 2, 8, 6, 2, 19, 2, 8, 2, 6, 2, 5, 6, 8, 2, 2, 8, 5, 22, 2, 6, 20, 2, 2, 9, 5, 23, 2, 9, 2, 5, 9, 7, 2, 5, 7, 9, 5, 2, 2, 25, 2, 16, 9, 2, 2, 21, 7, 2, 26, 5, 9, 5, 9, 7, 2, 7, 23, 2, 5, 10, 2

Offset

1,2

Comments

Let S_m be the sequence { k : rad(k) | rad(m) }. This sequence gives the number of k <= rad(m), which is the same as k <= m, since m is squarefree. Seen another way, this sequence gives the position of m in S_m.

Number of k <= p^e, e >= 0, such that rad(k) | p is (e+1). This is given by {A025477} + 1.

Number of k <= m for numbers k neither squarefree nor prime powers (k in A126706) is given by A365790(n) = A010846(A126706(n)).

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Formula

a(n) = A010846(A005117(n)).

Let b(n) = A005117(n).

For prime b(n) = p, a(n) = 2 since terms k in S_p such that k <= p are {1, p}.

For composite b(n) = m, a(n) > 2, since p | m appear in S_p, and p < m.

Example

Let b(n) = A005117(n).
a(1) = 1 since 1 is the only number k <= b(1) such that rad(k) | 1.
a(2) = 2 since k in {1, 2} are such that rad(k) | 2.
a(5) = 5 since b(5) = 6, k in {1, 2, 3, 4, 6} are such that rad(k) | 6. That is, 6 appears in the 5th position in S_6 = A003586.
a(7) = 6 since b(7) = 10, Card({ k : k <= 10, rad(k) | 10 }) = Card({1, 2, 4, 5, 8, 10}) = 6. That is, 10 appears in the 6th position in S_10 = A003592, etc.

Mathematica

rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Map[Function[{m, r}, Count[Range[m], _?(Divisible[r, rad[#] ] &)]] @@ {#, rad[#]} &, Select[Range[2^10], SquareFreeQ]]

Prog

(Python)
from math import gcd, isqrt
from sympy import mobius
def A363924(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return int(sum(mobius(k)*(m//k) for k in range(1, m+1) if gcd(m, k)==1)) # Chai Wah Wu, Aug 15 2024

Crossrefs

Cf. A003586, A003592, A005117, A007947, A010846, A025477, A126706, A365790.

Sequence in context: A368308 A080348 A355198 * A029662 A077913 A069862

Adjacent sequences: A363921 A363922 A363923 * A363925 A363926 A363927

Keyword

nonn

Author

Michael De Vlieger, Oct 24 2023

Status

approved