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.
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
REFERENCES
P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique (1980).
LINKS
Rémy Sigrist, Table of n, a(n) for n = 2..10000
Thomas Bloom, Problem #459, Erdős Problems.
Rémy Sigrist, PARI program for A289280
Rémy Sigrist, Logarithmic scatterplot of the first 100000 terms
FORMULA
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 *)
(* Second program *)
t=Table[Transpose[FactorInteger[n]][[1]], {n, 1, 40000}];
a[n_]:=SelectFirst[t[[n+1;; ]], SubsetQ[t[[n]], #]&->"Index"]+n;
Table[a[n], {n, 2, 200}] (* Elijah Beregovsky, Dec 28 2025 *)
PROG
(PARI) \\ See Links section.
CROSSREFS
KEYWORD
nonn,changed
AUTHOR
Rémy Sigrist, Jul 01 2017
EXTENSIONS
Revised by Elijah Beregovsky, Jan 02 2026
STATUS
approved