A326844 - OEIS

%I #10 Feb 10 2023 17:10:48

%S 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 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 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 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 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%H Antti Karttunen, <a href="/A326844/b326844.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = A001222(n) * A061395(n) - A056239(n).

%e The partition with Heinz number 7865 is (6,5,5,3), with diagram:

%e   o o o o o o

%e   o o o o o .

%e   o o o o o .

%e   o o o . . .

%e The size of the complement (shown in dots) in a 6 X 4 rectangle is 5, so a(7865) = 5.

%t 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 (PARI)

%o A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));

%o A061395(n) = if(n>1, primepi(vecmax(factor(n)[, 1])), 0);

%o A326844(n) = ((bigomega(n)*A061395(n)) - A056239(n)); \\ _Antti Karttunen_, Feb 10 2023

%Y Cf. A056239, A061395, A106529, A112798, A268192.

%Y Cf. A316413, A326836, A326837, A326845, A326846, A326848.

%K nonn

%O 1,10

%A _Gus Wiseman_, Jul 26 2019

%E Data section extended up to term a(100) by _Antti Karttunen_, Feb 10 2023