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A304720 Number of nonnegative integers k such that n - (4^k - k) is positive and squarefree.

%I A304720 #23 May 24 2018 02:40:06
%S A304720 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,
%T A304720 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,
%U A304720 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
%N A304720 Number of nonnegative integers k such that n - (4^k - k) is positive and squarefree.
%C A304720 Conjecture: a(n) > 0 for all n > 1.
%C A304720 This has been verified for n up to 2*10^10.
%C A304720 See A304721 for the values of n with a(n) = 1.
%C A304720 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
%H A304720 Zhi-Wei Sun, <a href="/A304720/b304720.txt">Table of n, a(n) for n = 1..100000</a>
%H A304720 Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/116f.pdf">Mixed sums of primes and other terms</a>, in: D. Chudnovsky and G. Chudnovsky (eds.), Additive Number Theory, Springer, New York, 2010, pp. 341-353.
%H A304720 Zhi-Wei Sun, <a href="https://doi.org/10.1007/978-3-319-68032-3_20">Conjectures on representations involving primes</a>, 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 <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>, 2012-2017.)
%e A304720 a(2) = 1 with 2 - (4^0 - 0) = 1 squarefree.
%e A304720 a(178) = 1 with 178 - (4^0 - 0) = 3*59 squarefree.
%e A304720 a(245) = 1 with 245 - (4^2 - 2) = 3*7*11 squarefree.
%e A304720 a(9196727) = 1 with 9196727 - (4^6 - 6) = 19*211*2293 squarefree.
%e A304720 a(16130577) = 1 with 16130577 - (4^9 - 9) = 2*7934221 squarefree.
%e A304720 a(38029402) = 1 with 38029402 - (4^1 - 1) = 1153*32983 squarefree.
%e A304720 a(180196927) = 1 with 180196927 - (4^11 - 11) = 2*139*227*2789 squarefree.
%t A304720 f[n_]:=f[n]=4^n-n;
%t A304720 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]
%Y A304720 Cf. A000302, A005117, A024037, A304034, A304081, A304331, A304333, A304522, A304523, A304689, A304721, A304943, A304945.
%Y A304720 Cf. A281192.
%K A304720 nonn
%O A304720 1,4
%A A304720 _Zhi-Wei Sun_, May 17 2018