cp's OEIS Frontend

A346699 Sum of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.

%I A346699 #11 Aug 09 2021 11:21:07
%S A346699 0,1,2,1,3,2,4,2,2,3,5,3,6,4,3,2,7,3,8,4,4,5,9,3,3,6,4,5,10,4,11,3,5,
%T A346699 7,4,3,12,8,6,4,13,5,14,6,5,9,15,4,4,4,7,7,16,4,5,5,8,10,17,4,18,11,6,
%U A346699 3,6,6,19,8,9,5,20,4,21,12,5,9,5,7,22,5,4
%N A346699 Sum of the odd bisection (odd-indexed parts) of the integer partition with Heinz number n.
%C A346699 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 A346699 a(n) = A056239(A346701(n)).
%F A346699 a(n) = A056239(n) - A346700(n).
%F A346699 a(n) = A344616(n) + A346700(n).
%F A346699 a(n odd omega) = A346697(n).
%F A346699 a(n even omega) = A346698(n).
%F A346699 A316524(n) = A346697(n) - A346698(n).
%e A346699 The partition with Heinz number 1100 is (5,3,3,1,1), so a(1100) = 5 + 3 + 1 = 9.
%e A346699 The partition with Heinz number 2100 is (4,3,3,2,1,1), so a(2100) = 4 + 3 + 1 = 8.
%t A346699 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A346699 Table[Total[First/@Partition[Append[Reverse[primeMS[n]],0],2]],{n,100}]
%Y A346699 The version for standard compositions is A209281(n+1) (even: A346633).
%Y A346699 Subtracting the even version gives A344616 (non-reverse: A316524).
%Y A346699 The even version is A346700.
%Y A346699 The non-reverse version (multisets instead of partitions) is A346697.
%Y A346699 The even non-reverse version is A346698.
%Y A346699 A001414 adds up prime factors, row sums of A027746.
%Y A346699 A056239 adds up prime indices, row sums of A112798.
%Y A346699 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A346699 A325534 counts separable partitions, ranked by A335433.
%Y A346699 A325535 counts inseparable partitions, ranked by A335448.
%Y A346699 A344606 counts alternating permutations of prime indices.
%Y A346699 Cf. A000070, A025047, A120452, A124754, A329888, A341446, A344617, A345957, A345958, A346701, A346703.
%K A346699 nonn
%O A346699 1,3
%A A346699 _Gus Wiseman_, Aug 03 2021