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A289280 a(n) = least integer k > n such that any prime factor of k is also a prime factor of n. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,1

COMMENTS

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),

  then n/lpf(n) < A001221(n) < n,

  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

LINKS

Rémy Sigrist, Table of n, a(n) for n = 2..10000

Rémy Sigrist, PARI program for A289280

Rémy Sigrist, Scatterplot of the ordinal transform of the first 100000 terms

Rémy Sigrist, Logarithmic scatterplot of the first 100000 terms

EXAMPLE

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.

From Michael De Vlieger, Jul 02 2017: (Start)

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)

MATHEMATICA

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 *)

PROG

See Links section.

CROSSREFS

Cf. A001221, A007947, A020639, A079277.

Sequence in context: A316346 A135718 A140580 * A077662 A063718 A063748

Adjacent sequences:  A289277 A289278 A289279 * A289281 A289282 A289283

KEYWORD

nonn

AUTHOR

Rémy Sigrist, Jul 01 2017

STATUS

approved

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Last modified October 16 10:19 EDT 2021. Contains 348041 sequences. (Running on oeis4.)