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A326844 Let y be the integer partition with Heinz number n. Then a(n) is the size of the complement, in the minimal rectangular partition containing the Young diagram of y, of the Young diagram of y.

%I A326844 #10 Feb 10 2023 17:10:48
%S A326844 0,0,0,0,0,1,0,0,0,2,0,2,0,3,1,0,0,1,0,4,2,4,0,3,0,5,0,6,0,3,0,0,3,6,
%T A326844 1,2,0,7,4,6,0,5,0,8,2,8,0,4,0,2,5,10,0,1,2,9,6,9,0,5,0,10,4,0,3,7,0,
%U A326844 12,7,4,0,3,0,11,1,14,1,9,0,8,0,12,0,8,4,13,8,12,0,4,2,16,9,14,5,5,0,3,6,4
%N A326844 Let y be the integer partition with Heinz number n. Then a(n) is the size of the complement, in the minimal rectangular partition containing the Young diagram of y, of the Young diagram of y.
%C A326844 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A326844 Antti Karttunen, <a href="/A326844/b326844.txt">Table of n, a(n) for n = 1..65537</a>
%F A326844 a(n) = A001222(n) * A061395(n) - A056239(n).
%e A326844 The partition with Heinz number 7865 is (6,5,5,3), with diagram:
%e A326844   o o o o o o
%e A326844   o o o o o .
%e A326844   o o o o o .
%e A326844   o o o . . .
%e A326844 The size of the complement (shown in dots) in a 6 X 4 rectangle is 5, so a(7865) = 5.
%t A326844 Table[If[n==1,0,With[{y=Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]},Max[y]*Length[y]-Total[y]]],{n,100}]
%o A326844 (PARI)
%o A326844 A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
%o A326844 A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);
%o A326844 A326844(n) = ((bigomega(n)*A061395(n)) - A056239(n)); \\ _Antti Karttunen_, Feb 10 2023
%Y A326844 Cf. A056239, A061395, A106529, A112798, A268192.
%Y A326844 Cf. A316413, A326836, A326837, A326845, A326846, A326848.
%K A326844 nonn
%O A326844 1,10
%A A326844 _Gus Wiseman_, Jul 26 2019
%E A326844 Data section extended up to term a(100) by _Antti Karttunen_, Feb 10 2023