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A371735
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Maximal length of a set partition of the binary indices of n into blocks all having the same sum.
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4
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0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1
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OFFSET
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0,8
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COMMENTS
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A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
If a(n) = k then the binary indices of n (row n of A048793) are k-quanimous (counted by A371783).
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LINKS
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EXAMPLE
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The binary indices of 119 are {1,2,3,5,6,7}, and the set partitions into blocks with the same sum are:
{{1,7},{2,6},{3,5}}
{{1,5,6},{2,3,7}}
{{1,2,3,6},{5,7}}
{{1,2,3,5,6,7}}
So a(119) = 3.
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MATHEMATICA
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sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]& /@ sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Table[Max[Length/@Select[sps[bix[n]], SameQ@@Total/@#&]], {n, 0, 100}]
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CROSSREFS
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A version for factorizations is A371733.
Positions of terms > 1 are A371784.
A070939 gives length of binary expansion.
A326031 gives weight of the set-system with BII-number n.
A371783 counts k-quanimous partitions.
Cf. A006827, A038041, A096111, A279787, A305551, A321451, A321455, A322794, A326534, A371731, A371734.
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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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