A285734 a(1) = 0, and for n > 1, a(n) = the largest squarefree number x such that x < n-x, and n-x is also squarefree.

0, 1, 1, 2, 2, 3, 2, 3, 3, 5, 5, 6, 6, 7, 5, 6, 7, 7, 6, 10, 10, 11, 10, 11, 11, 13, 13, 14, 14, 15, 14, 15, 14, 17, 14, 17, 15, 19, 17, 19, 19, 21, 21, 22, 22, 23, 21, 22, 23, 21, 22, 26, 23, 23, 26, 26, 26, 29, 29, 30, 30, 31, 30, 31, 31, 33, 33, 34, 34, 35, 34, 35, 35, 37, 37, 38, 38, 39, 38, 39, 39, 41, 41, 42, 42, 43, 41, 42, 43, 43, 38, 46, 46, 47, 42

Offset

1,4

Comments

For n > 1, a(n) = the largest squarefree number x <= n/2 for which n-x 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 the Mathematics Stack Exchange 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 y-x is the least possible.

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 - A285735(n).

Prog

(Scheme)
(define (A285734 n) (if (= 1 n) 0 (let loop ((j 1) (k (- n 1)) (s 0)) (if (> j k) s (loop (+ 1 j) (- k 1) (max s (* j (A008966 j) (A008966 k))))))))
;; Much faster version:
(define (A285734 n) (if (= 1 n) 0 (let loop ((j (floor->exact (/ n 2)))) (if (and (= 1 (A008966 j)) (= 1 (A008966 (- n j)))) j (loop (- j 1))))))
(Python)
from sympy.ntheory.factor_ import core
def issquarefree(n): return core(n) == n
def a285734(n):
    if n==1: return 0
    j=n//2
    while True:
        if issquarefree(j) and issquarefree(n - j): return j
        else: j-=1
print([a285734(n) for n in range(1, 101)]) # Indranil Ghosh, May 02 2017
(PARI) a(n)=forstep(x=n\2, 1, -1, if(issquarefree(x) && issquarefree(n-x), return(x))); 0 \\ Charles R Greathouse IV, Nov 05 2017

Crossrefs

Cf. A005117, A008966, A071068, A285718, A285719, A285735, A285736, A286106, A286107.

Sequence in context: A057334 A048475 A204597 * A051698 A046773 A175402

Adjacent sequences: A285731 A285732 A285733 * A285735 A285736 A285737

Keyword

nonn

Author

Antti Karttunen, May 02 2017

Status

approved