

A285735


a(1) = 1, and for n > 1, a(n) = the least squarefree number x such that x > nx, and nx is also squarefree.


9



1, 1, 2, 2, 3, 3, 5, 5, 6, 5, 6, 6, 7, 7, 10, 10, 10, 11, 13, 10, 11, 11, 13, 13, 14, 13, 14, 14, 15, 15, 17, 17, 19, 17, 21, 19, 22, 19, 22, 21, 22, 21, 22, 22, 23, 23, 26, 26, 26, 29, 29, 26, 30, 31, 29, 30, 31, 29, 30, 30, 31, 31, 33, 33, 34, 33, 34, 34, 35, 35, 37, 37, 38, 37, 38, 38, 39, 39, 41, 41, 42, 41, 42, 42, 43, 43, 46, 46, 46, 47, 53, 46, 47, 47
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OFFSET

1,3


COMMENTS

For n > 1, a(n) = the least squarefree number x >= n/2 for which nx is also squarefree.
For any n > 1 there is at least one decomposition of n as a sum of two squarefree numbers (cf. A071068 and Math Stackexchange link). Of all pairs (x,y) of squarefree numbers for which x <= y and x+y = n, sequences A285734 and A285735 give the unique pair for which the difference yx is the least possible.


LINKS

Antti Karttunen, Table of n, a(n) for n = 1..10000
Math Stackexchange, Sums of square free numbers, is this conjecture equivalent to Goldbach's conjecture? (See especially the answer of Aryabhata)
K. Rogers, The Schnirelmann density of the squarefree integers, Proc. Amer. Math. Soc. 15 (1964), pp. 515516.


FORMULA

a(n) = n  A285734(n).


PROG

(Scheme) (define (A285735 n) ( n (A285734 n)))
(Python)
from sympy.ntheory.factor_ import core
def issquarefree(n): return core(n) == n
def a285734(n):
if n==1: return 0
j=int(n/2)
while True:
if issquarefree(j) and issquarefree(n  j): return j
else: j=1
def a285735(n): return n  a285734(n)
print [a285735(n) for n in xrange(1, 101)] # Indranil Ghosh, May 02 2017
(PARI) a(n)=for(x=(n+1)\2, n, if(issquarefree(x) && issquarefree(nx), return(x))); 1 \\ Charles R Greathouse IV, Nov 05 2017


CROSSREFS

Cf. A005117, A008966, A071068, A285718, A285719, A285734, A285736, A286106, A286107.
Sequence in context: A082524 A099961 A286107 * A038810 A178503 A211275
Adjacent sequences: A285732 A285733 A285734 * A285736 A285737 A285738


KEYWORD

nonn


AUTHOR

Antti Karttunen, May 02 2017


STATUS

approved



