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Number of perfect powers m <= n such that rad(m) | n, where rad = A007947.
1

%I #7 Oct 05 2024 10:58:27

%S 1,1,1,2,1,2,1,3,2,3,1,4,1,3,2,4,1,5,1,4,2,4,1,5,2,4,3,4,1,7,1,5,3,5,

%T 2,8,1,5,3,6,1,8,1,5,4,5,1,8,2,6,3,5,1,8,2,6,3,5,1,9,1,5,4,6,2,9,1,6,

%U 3,8,1,9,1,6,4,6,2,9,1,7,4,6,1,11,2,6,4

%N Number of perfect powers m <= n such that rad(m) | n, where rad = A007947.

%C Cardinality of the intersection of A001597 and row n of A162306.

%H Michael De Vlieger, <a href="/A376719/b376719.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael De Vlieger, <a href="/A376719/a376719.png">Log log scatterplot of a(n)</a>, n = 1..2^18.

%F a(p) = 1 for prime p.

%F a(p^k) = k.

%F a(n) >= A091050(n).

%F a(n) = A091050(n) for prime powers n (A000961).

%e Table showing perfect powers in row n of A162306 for n <= 36 such that a(n) > 1:

%e 4: {1, 4}

%e 6: {1, 4}

%e 8: {1, 4, 8}

%e 9: {1, 9}

%e 10: {1, 4, 8}

%e 12: {1, 4, 8, 9}

%e 14: {1, 4, 8}

%e 15: {1, 9}

%e 16: {1, 4, 8, 16}

%e 18: {1, 4, 8, 9, 16}

%e 20: {1, 4, 8, 16}

%e 30: {1, 4, 8, 9, 16, 25, 27}

%e 36: {1, 4, 8, 9, 16, 27, 32, 36}

%t Table[Which[PrimeQ[n], 0,

%t PrimePowerQ[n], FactorInteger[n][[1, -1]] - 1,

%t True, Count[Range[n],

%t _?(And[Divisible[n, Times @@ #[[All, 1]]],

%t GCD @@ #[[All, -1]] > 1] &@ FactorInteger[#]} &)] ], {n, 120}]

%Y Cf. A000961, A001597, A007947, A091050, A162306.

%K nonn

%O 1,4

%A _Michael De Vlieger_, Oct 02 2024