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A304720
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Number of nonnegative integers k such that n - (4^k - k) is positive and squarefree.
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4
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0, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 2, 3, 2, 2, 1, 3, 1, 2, 1, 3, 2, 1, 2, 1, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 1, 3, 1, 2, 2, 3, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 3, 4, 2, 3, 3, 3, 1, 2, 2, 4, 2, 2, 3, 3, 2, 2, 3, 3, 1, 3, 2, 4, 1, 3, 2, 4, 2, 3, 2, 3
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OFFSET
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1,4
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COMMENTS
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Conjecture: a(n) > 0 for all n > 1.
This has been verified for n up to 2*10^10.
See A304721 for the values of n with a(n) = 1.
See A281192 for N such that none of N - 1 or N + 1 is squarefree: then n = N + 2 is such that n - 1 and n - 3 are not squarefree, i.e., one cannot take k = 0 or k = 1 in the present definition, and k > 1 is required to satisfy the conjecture. - M. F. Hasler, May 23 2018
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LINKS
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Zhi-Wei Sun, Conjectures on representations involving primes, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also arXiv:1211.1588 [math.NT], 2012-2017.)
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EXAMPLE
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a(2) = 1 with 2 - (4^0 - 0) = 1 squarefree.
a(178) = 1 with 178 - (4^0 - 0) = 3*59 squarefree.
a(245) = 1 with 245 - (4^2 - 2) = 3*7*11 squarefree.
a(9196727) = 1 with 9196727 - (4^6 - 6) = 19*211*2293 squarefree.
a(16130577) = 1 with 16130577 - (4^9 - 9) = 2*7934221 squarefree.
a(38029402) = 1 with 38029402 - (4^1 - 1) = 1153*32983 squarefree.
a(180196927) = 1 with 180196927 - (4^11 - 11) = 2*139*227*2789 squarefree.
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MATHEMATICA
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f[n_]:=f[n]=4^n-n;
tab={}; Do[r=0; k=0; Label[bb]; If[f[k]>=n, Goto[aa]]; If[SquareFreeQ[n-f[k]], r=r+1]; k=k+1; Goto[bb]; Label[aa]; tab=Append[tab, r], {n, 1, 100}]; Print[tab]
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CROSSREFS
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Cf. A000302, A005117, A024037, A304034, A304081, A304331, A304333, A304522, A304523, A304689, A304721, A304943, A304945.
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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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