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A298882
a(1) = 1, and for any n > 1, if n is the k-th number with least prime factor p, then a(n) is the k-th number with greatest prime factor p.
1
1, 2, 3, 4, 5, 8, 7, 16, 6, 32, 11, 64, 13, 128, 9, 256, 17, 512, 19, 1024, 12, 2048, 23, 4096, 10, 8192, 18, 16384, 29, 32768, 31, 65536, 24, 131072, 15, 262144, 37, 524288, 27, 1048576, 41, 2097152, 43, 4194304, 36, 8388608, 47, 16777216, 14, 33554432, 48
OFFSET
1,2
COMMENTS
This sequence is a permutation of the natural numbers, with inverse A298268.
For any prime p and k > 0:
- if s_p(k) is the k-th p-smooth number and r_p(k) is the k-th p-rough number,
- then a(p * r_p(k)) = p * s_p(k),
- for example: a(11 * A008364(k)) = 11 * A051038(k).
FORMULA
a(1) = 1.
a(A083140(n, k)) = A125624(n, k) for any n > 0 and k > 0.
a(n) = A125624(A055396(n), A078898(n)) for any n > 1.
Empirically:
- a(n) = n iff n belongs to A046022,
- a(2 * k) = 2^k for any k > 0,
- a(p^2) = 2 * p for any prime p,
- a(p * q) = 3 * p for any pair of consecutive odd primes (p, q).
EXAMPLE
The first terms, alongside A020639(n), are:
n a(n) lpf(n)
-- ---- ------
1 1 1
2 2 2
3 3 3
4 4 2
5 5 5
6 8 2
7 7 7
8 16 2
9 6 3
10 32 2
11 11 11
12 64 2
13 13 13
14 128 2
15 9 3
16 256 2
17 17 17
18 512 2
19 19 19
20 1024 2
CROSSREFS
KEYWORD
nonn
AUTHOR
Rémy Sigrist, Jan 28 2018
STATUS
approved