A285718 a(1) = 0, and for n > 1, a(n) = the least squarefree number x such that n-x is also squarefree.

0, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 3, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 3, 5, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 3, 6, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1

Offset

1,5

Comments

For any n > 1 there is at least one decomposition of n as a sum of two squarefree numbers (cf. A071068 and the Mathematics Stack Exchange 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 y-x is the largest possible.

Question: Are there arbitrarily large terms in this sequence?

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Mathematics Stack Exchange, 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. 515-516.

Formula

a(n) = n - A285719(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) = 5.

Mathematica

Table[If[n == 1, 0, x = 1; While[Nand[SquareFreeQ@ x, SquareFreeQ[n - x]], x++]; x], {n, 120}] (* Michael De Vlieger, May 03 2017 *)

Prog

(Scheme)
(define (A285718 n) (if (= 1 n) 0 (let loop ((k 1)) (if (not (zero? (A008683 (- n (A005117 k))))) (A005117 k) (loop (+ 1 k))))))
(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
print([a285718(n) for n in range(1, 121)]) # Indranil Ghosh, May 02 2017

Crossrefs

Cf. A005117, A008683, A013929, A071068, A285719, A285720, A285734.

Sequence in context: A340811 A340812 A228349 * A205792 A370784 A249739

Adjacent sequences: A285715 A285716 A285717 * A285719 A285720 A285721

Keyword

nonn

Author

Antti Karttunen, May 02 2017

Status

approved