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A289280
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a(n) = least integer k > n such that any prime factor of k is also a prime factor of n.
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9
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4, 9, 8, 25, 8, 49, 16, 27, 16, 121, 16, 169, 16, 25, 32, 289, 24, 361, 25, 27, 32, 529, 27, 125, 32, 81, 32, 841, 32, 961, 64, 81, 64, 49, 48, 1369, 64, 81, 50, 1681, 48, 1849, 64, 75, 64, 2209, 54, 343, 64, 81, 64, 2809, 64, 121, 64, 81, 64, 3481, 64, 3721
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OFFSET
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2,1
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COMMENTS
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In other words:
- a(n) is the least k > n such that rad(k) divides rad(n), where rad = A007947,
- or, if P_n denotes the set of prime factors of n, then a(n) is the least P_n-smooth number > n.
For any n > 1, n < a(n) <= n*lpf(n), where lpf = A020639.
a(p^k) = p^(k+1) for any prime p and k > 0.
a(n) is never squarefree.
This sequence has connections with A079277:
- here we search the least P_n-smooth number > n, there the largest < n,
- also, if omega(n) > 1 (where omega = A001221),
so n < A001221(n)*lpf(n) < n*lpf(n),
as A001221(n)*lpf(n) is P_n-smooth,
we have a(n) <= A001221(n)*lpf(n) < n*lpf(n),
and n cannot divide a(n).
The (logarithmic) scatterplot of the sequence has horizontal rays similar to those observed for A079277; they correspond to frequent values, typically with a small number of distinct prime divisors (see also scatterplots in Links section).
Given n < a(n) <= n*lpf(n), a(n) | n^m with m >= 2. Values of m: {2, 2, 2, 2, 3, 2, 2, 2, 4, 2, 2, 2, 4, 2, 2, 2, 3, 2, 2, 3, 5, 2, 3, 2, ...}. - Michael De Vlieger, Jul 02 2017
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LINKS
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EXAMPLE
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For n = 42:
- 42 = 2 * 3 * 7, hence P_42 = { 2, 3, 7 },
- the P_42-smooth numbers are: 1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 16, 18, 21, 24, 27, 28, 32, 36, 42, 48, 49, ...
- hence a(42) = 48.
a(n) divides n^m with m >= 2:
n a(n) m
2 4 2
3 9 2
4 8 2
5 25 2
6 8 3
7 49 2
8 16 2
9 27 2
10 16 4
11 121 2
12 16 2
13 169 2
14 16 4
15 25 2
16 32 2
17 289 2
18 24 3
19 361 2
20 25 2
(End)
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MATHEMATICA
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Table[Which[PrimeQ@ n, n^2, PrimePowerQ@ n, Block[{p = 2, e}, While[Set[e, IntegerExponent[n, p]] == 0, p = NextPrime@ p]; p^(e + 1)], True, Block[{k = n + 1}, While[PowerMod[n, k, k] != 0, k++]; k]], {n, 2, 61}] (* Michael De Vlieger, Jul 02 2017 *)
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PROG
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See Links section.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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