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A325135 Size of the integer partition with Heinz number n after its inner lining, or, equivalently, its largest hook, is removed.

%I A325135 #9 Apr 19 2019 11:21:41
%S A325135 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 A325135 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 A325135 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 A325135 Size of the integer partition with Heinz number n after its inner lining, or, equivalently, its largest hook, is removed.
%C A325135 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A325135 Antti Karttunen, <a href="/A325135/b325135.txt">Table of n, a(n) for n = 1..20000</a>
%H A325135 Antti Karttunen, <a href="/A325135/a325135.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A325135 a(n) = A056239(A325133(n)).
%F A325135 For n > 1:
%F A325135 a(n) = A056239(n) - A001222(n) - A061395(n) + 1.
%F A325135 a(n) = A056239(n) - A252464(n).
%F A325135 a(n) = A056239(n) - A325134(n) + 1.
%e A325135 The partition with Heinz number 715 is (6,5,3), with diagram
%e A325135   o o o o o o
%e A325135   o o o o o
%e A325135   o o o
%e A325135 which has inner lining
%e A325135           o o
%e A325135       o o o
%e A325135   o o o
%e A325135 or largest hook
%e A325135   o o o o o o
%e A325135   o
%e A325135   o
%e A325135 both of which have complement
%e A325135   o o o o
%e A325135   o o
%e A325135 which has size 6, so a(715) = 6.
%t A325135 Table[If[n==1,0,Total[Most[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]-1]],{n,100}]
%o A325135 (PARI)
%o A325135 A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
%o A325135 A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
%o A325135 A325135(n) = if(1==n,0,(1+A056239(n)-bigomega(n)-A061395(n))); \\ _Antti Karttunen_, Apr 14 2019
%Y A325135 Cf. A000720, A001222, A052126, A056239, A061395, A093641, A112798, A252464, A257990, A325133, A325134.
%K A325135 nonn
%O A325135 1,25
%A A325135 _Gus Wiseman_, Apr 02 2019
%E A325135 More terms from _Antti Karttunen_, Apr 14 2019