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A339662
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Greatest gap in the partition with Heinz number n.
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12
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0, 0, 1, 0, 2, 0, 3, 0, 1, 2, 4, 0, 5, 3, 1, 0, 6, 0, 7, 2, 3, 4, 8, 0, 2, 5, 1, 3, 9, 0, 10, 0, 4, 6, 2, 0, 11, 7, 5, 2, 12, 3, 13, 4, 1, 8, 14, 0, 3, 2, 6, 5, 15, 0, 4, 3, 7, 9, 16, 0, 17, 10, 3, 0, 5, 4, 18, 6, 8, 2, 19, 0, 20, 11, 1, 7, 3, 5, 21, 2, 1, 12
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OFFSET
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1,5
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COMMENTS
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We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
Also the index of the greatest prime, up to the greatest prime index of n, not dividing n. A prime index of n is a number m such that prime(m) divides n.
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LINKS
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FORMULA
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
maxgap[q_]:=Max@@Complement[Range[0, If[q=={}, 0, Max[q]]], q];
Table[maxgap[primeMS[n]], {n, 100}]
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CROSSREFS
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Positions of first appearances are A000040.
The version for positions of 1's in reversed binary expansion is A063250.
The prime itself (not just the index) is A079068.
The minimal instead of maximal version is A257993.
Positive integers by Heinz weight and image are counted by A339737.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A073491 lists numbers with gap-free prime indices.
Cf. A001223, A001522, A005117, A018818, A029707, A064391, A098743, A264401, A325351, A333214, A342192.
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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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