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%I #9 Apr 19 2019 11:21:34
%S 1,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,1,2,1,1,2,1,1,1,3,1,4,1,1,2,1,1,2,1,
%T 3,2,1,1,2,1,1,2,1,1,4,1,1,1,5,3,2,1,1,4,3,1,2,1,1,2,1,1,4,1,3,2,1,1,
%U 2,3,1,2,1,1,6,1,5,2,1,1,8,1,1,2,3,1,2,1,1,4,5,1,2,1,3,1,1,5,4,3,1,2,1,1,6
%N Heinz number of the integer partition obtained by removing the inner lining, or, equivalently, the largest hook, of the integer partition with Heinz number n.
%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H Antti Karttunen, <a href="/A325133/b325133.txt">Table of n, a(n) for n = 1..20000</a>
%H Antti Karttunen, <a href="/A325133/a325133.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F a(n) = A064989(A052126(n)) = A052126(A064989(n)).
%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 is the partition (4,2) with Heinz number 21, so a(715) = 21.
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[If[n==1,1,Times@@Prime/@DeleteCases[Most[primeMS[n]]-1,0]],{n,100}]
%o (PARI)
%o A052126(n) = if(1==n,n,n/vecmax(factor(n)[, 1]));
%o A064989(n) = { my(f); f = factor(n); if((n>1 && f[1,1]==2), f[1,2] = 0); for (i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f) };
%o A325133(n) = A052126(A064989(n)); \\ _Antti Karttunen_, Apr 14 2019
%Y Positions of ones are A093641 (Heinz numbers of hooks). The number of iterations required to reach 1 starting with n is A257990(n).
%Y Cf. A000720, A001222, A046660, A052126, A056239, A061395, A064989, A112798, A243055, A252464, A325134, A325135.
%K nonn
%O 1,9
%A _Gus Wiseman_, Apr 02 2019
%E More terms from _Antti Karttunen_, Apr 14 2019
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