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 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. 9
 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 (list; graph; refs; listen; history; text; internal format)
 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 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 y-x 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. 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

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Last modified August 14 03:43 EDT 2022. Contains 356110 sequences. (Running on oeis4.)