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A346700 Sum of the even bisection (even-indexed parts) of the integer partition with Heinz number n.

%I A346700 #14 Oct 15 2021 21:36:02
%S A346700 0,0,0,1,0,1,0,1,2,1,0,1,0,1,2,2,0,2,0,1,2,1,0,2,3,1,2,1,0,2,0,2,2,1,
%T A346700 3,3,0,1,2,2,0,2,0,1,2,1,0,2,4,3,2,1,0,3,3,2,2,1,0,3,0,1,2,3,3,2,0,1,
%U A346700 2,3,0,3,0,1,3,1,4,2,0,2,4,1,0,3,3,1,2,2,0,3,4,1,2,1,3,3,0,4,2,4,0,2,0,2,3
%N A346700 Sum of the even bisection (even-indexed parts) of the integer partition with Heinz number n.
%C A346700 First differs from A334107 at a(64) = 3, A334107(64) = 2.
%C A346700 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.
%H A346700 Antti Karttunen, <a href="/A346700/b346700.txt">Table of n, a(n) for n = 1..10201</a>
%H A346700 Antti Karttunen, <a href="/A346700/a346700.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%H A346700 <a href="/index/He#Heinz">Index entries for sequences related to Heinz numbers</a>
%F A346700 a(n) = A056239(n) - A346699(n).
%F A346700 a(n) = A346699(n) - A344616(n).
%F A346700 a(n even omega) = A346697(n).
%F A346700 a(n odd omega) = A346698(n).
%F A346700 A316524(n) = A346697(n) - A346698(n).
%F A346700 a(n) = A056239(A329888(n)). - _Gus Wiseman_ and _Antti Karttunen_, Oct 13 2021
%e A346700 The partition with Heinz number 1100 is (5,3,3,1,1), so a(1100) = 3 + 1 = 4.
%e A346700 The partition with Heinz number 2100 is (4,3,3,2,1,1), so a(2100) = 3 + 2 + 1 = 6.
%t A346700 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A346700 Table[Total[Last/@Partition[Reverse[primeMS[n]],2]],{n,100}]
%o A346700 (PARI) A346700(n) = if(1==n,0,my(f=factor(n),s=0,p=0); forstep(k=#f~,1,-1,while(f[k,2], s += (p%2)*primepi(f[k,1]); f[k,2]--; p++)); (s)); \\ _Antti Karttunen_, Sep 21 2021
%Y A346700 Sum of prime indices of A329888(n).
%Y A346700 Subtracting from the odd version gives A344616 (non-reverse: A316524).
%Y A346700 The unreversed version for standard compositions is A346633.
%Y A346700 The odd non-reverse version is A346697.
%Y A346700 The non-reverse version (multisets instead of partitions) is A346698.
%Y A346700 The odd version is A346699.
%Y A346700 A001414 adds up prime factors, row sums of A027746.
%Y A346700 A056239 adds up prime indices, row sums of A112798.
%Y A346700 A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A346700 A325534 counts separable partitions, ranked by A335433.
%Y A346700 A325535 counts inseparable partitions, ranked by A335448.
%Y A346700 A344606 counts alternating permutations of prime indices.
%Y A346700 Cf. A000070, A025047, A124754, A209281, A329888, A341446, A344617, A344653, A345957, A345958, A346701, A346702, A346704.
%K A346700 nonn
%O A346700 1,9
%A A346700 _Gus Wiseman_, Aug 03 2021
%E A346700 Data section extended up to 105 terms by _Antti Karttunen_, Sep 21 2021