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A285719
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a(1) = 1, and for n > 1, a(n) = the largest squarefree number k such that n-k is also squarefree.
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4
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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
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1,3
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COMMENTS
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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.
Note: a(n+1) differs from A070321(n) for the first time at n=50, with a(51) = 46, while A070321(50) = 47.
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LINKS
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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lsfn[n_]:=Module[{k=n-1}, While[!SquareFreeQ[k]||!SquareFreeQ[n-k], k--]; k]; Join[{1}, Array[ lsfn, 100, 2]] (* Harvey P. Dale, Apr 27 2023 *)
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PROG
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(Scheme)
(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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