login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A285719 a(1) = 1, and for n > 1, a(n) = the largest squarefree number k such that n-k 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 (list; graph; refs; listen; history; text; internal format)
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 y-x 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. 515-516.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified August 3 20:28 EDT 2021. Contains 346441 sequences. (Running on oeis4.)