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A346701 Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

%I A346701 #13 Oct 27 2021 22:23:34
%S A346701 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 A346701 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 A346701 9,11,14,19,29,59,10,61,31,21,8,13,22,67,34,23,14,71
%N A346701 Heinz number of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.
%C A346701 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 A346701 a(n) * A329888(n) = n.
%F A346701 A056239(a(n)) = A346699(n).
%e A346701 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 A346701 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 A346701 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A346701 Table[Times@@Prime/@First/@Partition[Append[Reverse[primeMS[n]],0],2],{n,100}]
%Y A346701 Positions of last appearances are A000290 without the first term 0.
%Y A346701 Positions of primes are A037143 (complement: A033942).
%Y A346701 The even version is A329888.
%Y A346701 Positions of first appearances are A342768.
%Y A346701 The sum of prime indices of a(n) is A346699(n), non-reverse: A346697.
%Y A346701 The non-reverse version is A346703.
%Y A346701 The even non-reverse version is A346704.
%Y A346701 A001221 counts distinct prime factors.
%Y A346701 A001222 counts all prime factors.
%Y A346701 A056239 adds up prime indices, row sums of A112798.
%Y A346701 A103919 counts partitions by sum and alternating sum, reverse A344612.
%Y A346701 A209281 (shifted) adds up the odd bisection of standard compositions.
%Y A346701 A316524 gives the alternating sum of prime indices, reverse A344616.
%Y A346701 A325534 counts separable partitions, ranked by A335433.
%Y A346701 A325535 counts inseparable partitions, ranked by A335448.
%Y A346701 A344606 counts alternating permutations of prime indices.
%Y A346701 A344617 gives the sign of the alternating sum of prime indices.
%Y A346701 A346700 gives the sum of the even bisection of reversed prime indices.
%Y A346701 Cf. A025047, A027187, A027193, A053738, A106356, A277103, A341446, A344653, A345957, A345958, A345959, A346698, A346702.
%K A346701 nonn
%O A346701 1,2
%A A346701 _Gus Wiseman_, Aug 03 2021