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A376719
Number of perfect powers m <= n such that rad(m) | n, where rad = A007947.
1
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, 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, 3, 8, 1, 9, 1, 6, 4, 6, 2, 9, 1, 7, 4, 6, 1, 11, 2, 6, 4
OFFSET
1,4
COMMENTS
Cardinality of the intersection of A001597 and row n of A162306.
LINKS
Michael De Vlieger, Log log scatterplot of a(n), n = 1..2^18.
FORMULA
a(p) = 1 for prime p.
a(p^k) = k.
a(n) >= A091050(n).
a(n) = A091050(n) for prime powers n (A000961).
EXAMPLE
Table showing perfect powers in row n of A162306 for n <= 36 such that a(n) > 1:
4: {1, 4}
6: {1, 4}
8: {1, 4, 8}
9: {1, 9}
10: {1, 4, 8}
12: {1, 4, 8, 9}
14: {1, 4, 8}
15: {1, 9}
16: {1, 4, 8, 16}
18: {1, 4, 8, 9, 16}
20: {1, 4, 8, 16}
30: {1, 4, 8, 9, 16, 25, 27}
36: {1, 4, 8, 9, 16, 27, 32, 36}
MATHEMATICA
Table[Which[PrimeQ[n], 0,
PrimePowerQ[n], FactorInteger[n][[1, -1]] - 1,
True, Count[Range[n],
_?(And[Divisible[n, Times @@ #[[All, 1]]],
GCD @@ #[[All, -1]] > 1] &@ FactorInteger[#]} &)] ], {n, 120}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael De Vlieger, Oct 02 2024
STATUS
approved