OFFSET
1,1
COMMENTS
If p = A006530(a(n)) then p * a(n) is in the sequence. E.g., as 12 is in the sequence with gpf(12) = A006530(12) = 3, 12*3^k is in the sequence for k > 0. Conjecture: if m is in the sequence then so is A003961(m). - David A. Corneth, Jul 13 2017
At present this and A027855 are complements in the set of integers >= 2. If a 1 were inserted at the start, then this and A027855 are complements in the set of positive integers. - Harry Richman, Sep 08 2019
The sequence is closed under multiplication (a semigroup). For, suppose x = p^i*m1, y = q^j*m2 are in the sequence, with p, q, p^i, p^j as given, with m1 > p and m2 > q, and suppose q >= p. If q = p then xy/q^(i+j) = m1*m2 > q. If q > p, then xy/q^j = p^i*m1*m2 > q (since q > p and p is greater than all primes in m1). - Richard Peterson, May 29 2022
There are subsequences that constitute subsemigroups: Consider as a subsequence all terms x such that x/p^k > a*p^b, with p,k as specified in the definition and a,b fixed real numbers greater than or equal to 1. Each pair (a,b) determines a subsequence that is also a subsemigroup of the original (1,1) semigroup that constitutes the whole sequence. The proof of closure is similar. To see that such proposed subsemigroups are nonempty, choose any prime p greater than 2 and multiply p by a sufficiently large power of 2. - Richard Peterson, May 29 2022
This sequence is a subsequence and subsemigroup of A289484. - Richard Peterson, Oct 29 2022
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..10000
EXAMPLE
From Michael De Vlieger, Jul 13 2017: (Start)
(End)
MATHEMATICA
Select[Range@ 280, Function[n, (n/Apply[Power, Last@ #]) > #[[-1, 1]] &@ FactorInteger[n]]] (* Michael De Vlieger, Jul 13 2017 *)
PROG
(PARI) isok(n) = {my(f = factor(n)); my(maxf = #f~); my(p = f[maxf, 1]); my(pk = f[maxf, 2]); (n/p^pk) > p; } \\ Michel Marcus, Jan 16 2014
(Python)
from sympy import factorint, primefactors
def a053585(n):
if n==1: return 1
p = primefactors(n)[-1]
return p**factorint(n)[p]
print([n for n in range(2, 301) if n>a053585(n)*primefactors(n)[-1]]) # Indranil Ghosh, Jul 13 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Extended by Ray Chandler, Nov 17 2008
Offset changed to 1 by Michel Marcus, Jan 16 2014
STATUS
approved