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A365533
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a(n) is the nim-value of the SALIQUANT game where the option is to subtract a nondivisor from 2*n.
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0
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0, 1, 1, 3, 2, 4, 6, 7, 4, 7, 5, 10, 12, 10, 13, 15, 8, 13, 9, 17, 17, 16, 11, 22, 22, 19, 25, 24, 14, 22, 30, 31, 16, 33, 32, 31, 18, 28, 19, 37, 20, 38, 21, 38, 37, 34, 23, 46, 45, 37, 42, 51, 26, 40, 27, 52, 28, 43, 29, 52, 60, 61, 52, 63, 58, 49, 66, 59, 34, 52, 35, 67, 36, 55, 62
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OFFSET
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1,4
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COMMENTS
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a(n) = SG(2*n) where SG(n) = mex{x in opt(n)} SG(x), where mex(A) is the least nonnegative integer not appearing in A, and opt(n) is a vector of values n-k where 1 <= k <= n is not a divisor of n. For odd n, SG(n) = (n-1)/2.
Note that SG(n) represents the nim-value (also called the Sprague-Grundy number) for position n in combinatorial game theory. It indicates the winning strategy for that position when both players play optimally.
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LINKS
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MATHEMATICA
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mex[l_List]:=Module[{i=0}, While[MemberQ[l, i], i++]; i]; SG[n_Integer?Positive]:=SG[n]=Module[{p, d}, If[n==1, Return[0]]; d=Select[Range[1, n], Mod[n, #]!=0&]; p=n-d; mex[SG[#]&/@p]]; a[n_]:=Module[{r={}}, Do[AppendTo[r, SG[2*i]], {i, n}]; r]; a[75] (* Robert P. P. McKone, Sep 09 2023 *)
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PROG
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(PARI) opt(n) = my(list=List()); for (k=1, n, if (n % k, listput(list, n-k))); Vec(vecsort(list));
lista(nn) = {nn *= 2; my(vsg = vector(nn, n, if (n%2, (n-1)/2))); forstep (n=2, nn, 2, my(v=row(n), list=List()); for (i=1, #v, listput(list, vsg[v[i]])); list = Vec(vecsort(list)); if (#list==0, vsg[n] = 0, for (k=1, vecmax(list)+1, if (!vecsearch(list, k), vsg[n] = k; break))); ); vector(nn\2, k, vsg[2*k]); }
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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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