A325224 Sum of prime indices of n minus the lesser of the number of prime factors of n counted with multiplicity and the maximum prime index of n.

0, 0, 1, 1, 2, 1, 3, 2, 2, 2, 4, 2, 5, 3, 3, 3, 6, 3, 7, 2, 4, 4, 8, 3, 4, 5, 4, 3, 9, 3, 10, 4, 5, 6, 5, 4, 11, 7, 6, 3, 12, 4, 13, 4, 4, 8, 14, 4, 6, 4, 7, 5, 15, 5, 6, 3, 8, 9, 16, 4, 17, 10, 5, 5, 7, 5, 18, 6, 9, 5, 19, 5, 20, 11, 5, 7, 7, 6, 21, 4, 6, 12

Offset: 1

Author

Gus Wiseman, Apr 12 2019

Keywords

• nonn

Comments

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

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

Examples

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

Links

• FindStat, St000533: The maximal number of non-attacking rooks on a Ferrers shape

Crossrefs

The number of times k appears in the sequence is A325232(k).

Cf. A001222, A052126, A056239, A061395, A064989, A065770, A112798, A174090, A257990, A263297, A325134, A325169, A325223, A325225, A325227.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[n==1,0,Total[primeMS[n]]-Min[Length[primeMS[n]],Max[primeMS[n]]]],{n,100}]

Formula

a(n) = A056239(n) - min(A001222(n), A061395(n)) = A056239(n) - A325225(n).