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Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.
9

%I #13 Oct 27 2021 22:23:34

%S 1,2,3,2,5,3,7,4,3,5,11,6,13,7,5,4,17,6,19,10,7,11,23,6,5,13,9,14,29,

%T 10,31,8,11,17,7,6,37,19,13,10,41,14,43,22,15,23,47,12,7,10,17,26,53,

%U 9,11,14,19,29,59,10,61,31,21,8,13,22,67,34,23,14,71

%N Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

%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.

%F a(n) * A329888(n) = n.

%F A056239(a(n)) = A346699(n).

%e The partition (2,2,2,1,1) has Heinz number 108 and odd bisection (2,2,1) with Heinz number 18, so a(108) = 18.

%e The partitions (3,2,2,1,1), (3,2,2,2,1), (3,3,2,1,1) have Heinz numbers 180, 270, 300 and all have odd bisection (3,2,1) with Heinz number 30, so a(180) = a(270) = a(300) = 30.

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

%t Table[Times@@Prime/@First/@Partition[Append[Reverse[primeMS[n]],0],2],{n,100}]

%Y Positions of last appearances are A000290 without the first term 0.

%Y Positions of primes are A037143 (complement: A033942).

%Y The even version is A329888.

%Y Positions of first appearances are A342768.

%Y The sum of prime indices of a(n) is A346699(n), non-reverse: A346697.

%Y The non-reverse version is A346703.

%Y The even non-reverse version is A346704.

%Y A001221 counts distinct prime factors.

%Y A001222 counts all prime factors.

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

%Y A103919 counts partitions by sum and alternating sum, reverse A344612.

%Y A209281 (shifted) adds up the odd bisection of standard compositions.

%Y A316524 gives the alternating sum of prime indices, reverse A344616.

%Y A325534 counts separable partitions, ranked by A335433.

%Y A325535 counts inseparable partitions, ranked by A335448.

%Y A344606 counts alternating permutations of prime indices.

%Y A344617 gives the sign of the alternating sum of prime indices.

%Y A346700 gives the sum of the even bisection of reversed prime indices.

%Y Cf. A025047, A027187, A027193, A053738, A106356, A277103, A341446, A344653, A345957, A345958, A345959, A346698, A346702.

%K nonn

%O 1,2

%A _Gus Wiseman_, Aug 03 2021