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
Keywords
Examples
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.
Links
- Michael De Vlieger, Table of n, a(n) for n = 1..10000
Programs
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Mathematica
rad[x_] := Times @@ FactorInteger[x][[All, 1]]; Map[Function[{m, r}, Count[Range[m], _?(Divisible[r, rad[#] ] &)]] @@ {#, rad[#]} &, Select[Range[2^10], SquareFreeQ]] -
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
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