

A285719


a(1) = 1, and for n > 1, a(n) = the largest squarefree number k such that nk is also squarefree.


4



1, 1, 2, 3, 3, 5, 6, 7, 7, 7, 10, 11, 11, 13, 14, 15, 15, 17, 17, 19, 19, 21, 22, 23, 23, 23, 26, 26, 26, 29, 30, 31, 31, 33, 34, 35, 35, 37, 38, 39, 39, 41, 42, 43, 43, 43, 46, 47, 47, 47, 46, 51, 51, 53, 53, 55, 55, 57, 58, 59, 59, 61, 62, 62, 62, 65, 66, 67, 67, 69, 70, 71, 71, 73, 74, 74, 74, 77, 78, 79, 79, 79, 82, 83, 83, 85, 86, 87, 87, 89, 89, 91, 91
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OFFSET

1,3


COMMENTS

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 positive squarefree numbers for which x <= y and x+y = n, sequences A285718 and A285719 give the unique pair for which the difference yx is the largest possible.
Note: a(n+1) differs from A070321(n) for the first time at n=50, with a(51) = 46, while A070321(50) = 47.


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  A285718(n).


EXAMPLE

For n=51 we see that 50 (2*5*5), 49 (7*7) and 48 (2^4 * 3) are all nonsquarefree (A013929). 47 (a prime) is squarefree, but 51  47 = 4 is not. On the other hand, both 46 (2*23) and 5 are squarefree numbers, thus a(51) = 46.


PROG

(Scheme)
(define (A285719 n) ( n (A285718 n)))
(define (A285719 n) (if (= 1 n) n (let loop ((k (A013928 n))) (if (not (zero? (A008683 ( n (A005117 k))))) (A005117 k) (loop ( k 1))))))
(Python)
from sympy.ntheory.factor_ import core
def issquarefree(n): return core(n) == n
def a285718(n):
if n==1: return 0
x = 1
while True:
if issquarefree(x) and issquarefree(n  x):return x
else: x+=1
def a285719(n): return n  a285718(n)
print([a285719(n) for n in range(1, 121)]) # Indranil Ghosh, May 02 2017


CROSSREFS

Cf. A005117, A008683, A013928, A013929, A070321, A071068, A285718, A285735.
Sequence in context: A081211 A081213 A081210 * A070321 A239904 A334819
Adjacent sequences: A285716 A285717 A285718 * A285720 A285721 A285722


KEYWORD

nonn


AUTHOR

Antti Karttunen, May 02 2017


STATUS

approved



