A325166 - OEIS

%I #10 Apr 19 2019 11:22:02

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

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

%H Antti Karttunen, <a href="/A325166/b325166.txt">Table of n, a(n) for n = 1..20000</a>

%H Antti Karttunen, <a href="/A325166/a325166.txt">Data supplement: n, a(n) computed for n = 1..65537</a>

%F a(n) = A056239(n) - A061395(n) - A001222(n) + A001221(n).

%F a(n) = A056239(n) - A297113(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 with internal portion

%e   o o o o o

%e   o o o o

%e   o o o

%e of size 12, so a(7865) = 12.

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[If[n==1,0,Total[primeMS[n]]-Max[primeMS[n]]-Length[primeMS[n]]+Length[Union[primeMS[n]]]],{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 A325166(n) = (A056239(n) - A061395(n) - bigomega(n) + omega(n)); \\ _Antti Karttunen_, Apr 14 2019

%Y Positions of zeros are A174090.

%Y Cf. A001221, A001222, A052126, A056239, A061395, A064989, A065770, A112798, A252464, A257990, A297113, A325133, A325135, A325167, A325169.

%K nonn

%O 1,15

%A _Gus Wiseman_, Apr 05 2019

%E More terms from _Antti Karttunen_, Apr 14 2019