%I #19 Apr 22 2021 01:42:43

%S 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,

%T 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,

%U 5,4,18,6,8,2,19,0,20,11,1,7,3,5,21,2,1,12

%N Greatest gap in the partition with Heinz number n.

%C We define the greatest gap of a partition to be the greatest nonnegative integer less than the greatest part and not in the partition.

%C 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.

%C 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.

%H George E. Andrews and David Newman, <a href="https://doi.org/10.1007/s00026-019-00427-w">Partitions and the Minimal Excludant</a>, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000474/">Dyson's crank of a partition</a>.

%H Brian Hopkins, James A. Sellers, and Dennis Stanton, <a href="https://arxiv.org/abs/2009.10873">Dyson's Crank and the Mex of Integer Partitions</a>, arXiv:2009.10873 [math.CO], 2020.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Mex_(mathematics)">Mex (mathematics)</a>

%F a(n) = A000720(A079068(n)).

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t maxgap[q_]:=Max@@Complement[Range[0,If[q=={},0,Max[q]]],q];

%t Table[maxgap[primeMS[n]],{n,100}]

%Y Positions of first appearances are A000040.

%Y Positions of 0's are A055932.

%Y The version for positions of 1's in reversed binary expansion is A063250.

%Y The prime itself (not just the index) is A079068.

%Y The version for crank is A257989.

%Y The minimal instead of maximal version is A257993.

%Y The version for greatest difference is A286469 or A286470.

%Y Positive integers by Heinz weight and image are counted by A339737.

%Y Positions of 1's are A339886.

%Y A000070 counts partitions with a selected part.

%Y A006128 counts partitions with a selected position.

%Y A015723 counts strict partitions with a selected part.

%Y A056239 adds up prime indices, row sums of A112798.

%Y A073491 lists numbers with gap-free prime indices.

%Y A238709/A238710 count partitions by least/greatest difference.

%Y A342050/A342051 have prime indices with odd/even least gap.

%Y Cf. A001223, A001522, A005117, A018818, A029707, A064391, A098743, A264401, A325351, A333214, A342192.

%K nonn

%O 1,5

%A _Gus Wiseman_, Apr 20 2021