cp's OEIS Frontend

A325166 Size of the internal portion of the integer partition with Heinz number n.

%I A325166 #10 Apr 19 2019 11:22:02
%S A325166 0,0,0,0,0,1,0,0,1,1,0,1,0,1,2,0,0,2,0,1,2,1,0,1,2,1,2,1,0,3,0,0,2,1,
%T A325166 3,2,0,1,2,1,0,3,0,1,3,1,0,1,3,3,2,1,0,3,3,1,2,1,0,3,0,1,3,0,3,3,0,1,
%U A325166 2,4,0,2,0,1,4,1,4,3,0,1,3,1,0,3,3,1,2,1,0,4,4,1,2,1,3,1,0,4,3,3,0,3,0,1,5
%N A325166 Size of the internal portion of the integer partition with Heinz number n.
%C A325166 The internal portion of an integer partition consists of all squares in the Young diagram that have a square both directly below and directly to the right.
%C A325166 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A325166 Antti Karttunen, <a href="/A325166/b325166.txt">Table of n, a(n) for n = 1..20000</a>
%H A325166 Antti Karttunen, <a href="/A325166/a325166.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A325166 a(n) = A056239(n) - A061395(n) - A001222(n) + A001221(n).
%F A325166 a(n) = A056239(n) - A297113(n).
%e A325166 The partition with Heinz number 7865 is (6,5,5,3), with diagram
%e A325166   o o o o o o
%e A325166   o o o o o
%e A325166   o o o o o
%e A325166   o o o
%e A325166 with internal portion
%e A325166   o o o o o
%e A325166   o o o o
%e A325166   o o o
%e A325166 of size 12, so a(7865) = 12.
%t A325166 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A325166 Table[If[n==1,0,Total[primeMS[n]]-Max[primeMS[n]]-Length[primeMS[n]]+Length[Union[primeMS[n]]]],{n,100}]
%o A325166 (PARI)
%o A325166 A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
%o A325166 A061395(n) = if(1==n, 0, primepi(vecmax(factor(n)[, 1])));
%o A325166 A325166(n) = (A056239(n) - A061395(n) - bigomega(n) + omega(n)); \\ _Antti Karttunen_, Apr 14 2019
%Y A325166 Positions of zeros are A174090.
%Y A325166 Cf. A001221, A001222, A052126, A056239, A061395, A064989, A065770, A112798, A252464, A257990, A297113, A325133, A325135, A325167, A325169.
%K A325166 nonn
%O A325166 1,15
%A A325166 _Gus Wiseman_, Apr 05 2019
%E A325166 More terms from _Antti Karttunen_, Apr 14 2019