A289280 a(n) = least integer k > n such that any prime factor of k is also a prime factor of n.

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

Offset: 2

Author

Rémy Sigrist, Jul 01 2017

Keywords

• nonn

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

Examples

			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)
		

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

Crossrefs

Cf. A001221, A007947, A020639, A079277.

Programs

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 *)

PARI

\\ See Links section.