%I #9 Apr 19 2019 11:21:41

%S 0,0,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0,1,0,0,0,2,0,2,0,0,1,0,0,1,0,

%T 2,1,0,0,1,0,0,1,0,0,2,0,0,0,3,2,1,0,0,2,2,0,1,0,0,1,0,0,2,0,2,1,0,0,

%U 1,2,0,1,0,0,3,0,3,1,0,0,3,0,0,1,2,0,1,0,0,2,3,0,1,0,2,0,0,3,2,2,0,1,0,0,3

%N Size of the integer partition with Heinz number n after its inner lining, or, equivalently, its largest hook, is removed.

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

%H Antti Karttunen, <a href="/A325135/b325135.txt">Table of n, a(n) for n = 1..20000</a>

%H Antti Karttunen, <a href="/A325135/a325135.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F a(n) = A056239(A325133(n)).

%F For n > 1:

%F a(n) = A056239(n) - A001222(n) - A061395(n) + 1.

%F a(n) = A056239(n) - A252464(n).

%F a(n) = A056239(n) - A325134(n) + 1.

%e The partition with Heinz number 715 is (6,5,3), with diagram

%e o o o o o o

%e o o o o o

%e o o o

%e which has inner lining

%e o o

%e o o o

%e o o o

%e or largest hook

%e o o o o o o

%e o

%e o

%e both of which have complement

%e o o o o

%e o o

%e which has size 6, so a(715) = 6.

%t Table[If[n==1,0,Total[Most[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]-1]],{n,100}]

%o (PARI)

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

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

%o A325135(n) = if(1==n,0,(1+A056239(n)-bigomega(n)-A061395(n))); \\ _Antti Karttunen_, Apr 14 2019

%Y Cf. A000720, A001222, A052126, A056239, A061395, A093641, A112798, A252464, A257990, A325133, A325134.

%K nonn

%O 1,25

%A _Gus Wiseman_, Apr 02 2019

%E More terms from _Antti Karttunen_, Apr 14 2019