A325223 - OEIS

%I #10 Apr 13 2019 08:11:01

%S 0,0,0,0,0,1,0,0,2,1,0,1,0,1,2,0,0,2,0,2,2,1,0,1,3,1,3,2,0,3,0,0,2,1,

%T 3,2,0,1,2,2,0,3,0,2,4,1,0,1,4,4,2,2,0,3,3,3,2,1,0,3,0,1,4,0,3,3,0,2,

%U 2,4,0,2,0,1,5,2,4,3,0,2,4,1,0,4,3,1,2

%N Sum of the prime indices of n minus the greater of the number of prime factors of n counted with multiplicity and the largest prime index of n.

%C A prime index of n is a number m such that prime(m) divides n.

%C Also the number of squares in the Young diagram of the integer partition with Heinz number n after the first row or the first column, whichever is larger, is removed. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

%H FindStat, <a href="http://www.findstat.org/StatisticsDatabase/St000784">St000784: The maximum of the length and the largest part of the integer partition</a>

%F a(n) = A056239(n) - max(A001222(n), A061395(n)) = A056239(n) - A263297(n).

%e 88 has 4 prime indices {1,1,1,5} with sum 8 and maximum 5, so a(88) = 8 - max(4,5) = 3.

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

%t Table[Total[primeMS[n]]-Max[Length[primeMS[n]],Max[primeMS[n]]],{n,100}]

%Y Positions of 0's are A174090. Positions of 1's are A325231.

%Y Cf. A001222, A051924, A052126, A056239, A061395, A064989, A065770, A112798, A257990, A263297, A325134, A325169, A325224, A325225, A325227.

%K nonn

%O 1,9

%A _Gus Wiseman_, Apr 12 2019