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A304721
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Numbers m with A304720(m) = 1.
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4
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2, 3, 5, 7, 9, 10, 11, 12, 13, 19, 21, 23, 26, 28, 30, 39, 41, 46, 50, 51, 53, 55, 57, 59, 77, 89, 93, 101, 113, 129, 149, 151, 153, 161, 165, 178, 185, 189, 201, 221, 237, 245, 246, 297, 364, 377, 489, 553, 581, 639
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OFFSET
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1,1
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COMMENTS
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Conjecture: The sequence only has 112 terms as listed in the b-file.
We have verified that there is no new term below 2*10^9.
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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(9) = 13 since 13 - (4^1 - 1) = 2*5 is squarefree, 13 - (4^0 - 0) = 2^2*3 is not squarefree, and 13 - (4^k -k ) < 0 for any integer k > 1.
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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]>=m, Goto[aa]]; If[SquareFreeQ[m-f[k]], r=r+1]; If[r>1, Goto[cc]]; k=k+1; Goto[bb]; Label[aa]; If[r==1, tab=Append[tab, m]]; Label[cc], {m, 1, 640}]; Print[tab]
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CROSSREFS
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Cf. A000302, A005117, A024037, A304034, A304081, A304331, A304333, A304522, A304523, A304689, A304720.
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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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